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 9781611972559, 2003042468

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Table of contents :
Title
Contents
List of Figures
Notation
Preface
1. Introduction to the Central Concepts
2. Convex Functions with Combinatorial Structures
3. Convex Analysis, Linear programming, and Integrality
4. M-Convex Sets and Submodular Set Functions
5. L-Convex Sets and Distance Functions
6. M-Convex Functions
7. L-Convex Functions
8. Conjugacy and Duality
9. Network Flows
10. Algorithms
11. Application to Mathematical Economics
12. Application to Systems Analysis by Mixed Matrices
Bibliography
Index

Citation preview

Discrete Convex Analysis

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Siam Monographs on Discrete Mathematics and Applications The series includes advanced monographs reporting on the most recent theoretical, computational, or applied developments in the field; introductory volumes aimed at mathematicians and other mathematically motivated readers interested in understanding certain areas of pure or applied combinatorics; and graduate textbooks. The volumes are devoted to various areas of discrete mathematics and its applications. Mathematicians, computer scientists, operations researchers, computationally oriented natural and social scientists, engineers, medical researchers, and other practitioners will find the volumes of interest. Editor-in-Chief Peter L. Hammer, RUTCOR, Rutgers, The State University of New Jersey Editorial Board M. Aigner, Freie Universität Berlin, Germany N. Alon, Tel Aviv University, Israel E. Balas, Carnegie Mellon University, USA J.- C. Bermond, Université de Nice–Sophia Antipolis, France J. Berstel, Université Marne-la-Vallée, France N. L. Biggs, The London School of Economics, United Kingdom B. Bollobás, University of Memphis, USA R. E. Burkard, Technische Universität Graz, Austria D. G. Corneil, University of Toronto, Canada I. Gessel, Brandeis University, USA F. Glover, University of Colorado, USA M. C. Golumbic, Bar-Ilan University, Israel R. L. Graham, AT&T Research, USA A. J. Hoffman, IBM T. J. Watson Research Center, USA T. Ibaraki, Kyoto University, Japan H. Imai, University of Tokyo, Japan M. Karon´ski, Adam Mickiewicz University, Poland, and Emory University, USA R. M. Karp, University of Washington, USA V. Klee, University of Washington, USA K. M. Koh, National University of Singapore, Republic of Singapore B. Korte, Universität Bonn, Germany

A. V. Kostochka, Siberian Branch of the Russian Academy of Sciences, Russia F. T. Leighton, Massachusetts Institute of Technology, USA T. Lengauer, Gesellschaft für Mathematik und Datenverarbeitung mbH, Germany S. Martello, DEIS University of Bologna, Italy M. Minoux, Université Pierre et Marie Curie, France R. Möhring, Technische Universität Berlin, Germany C. L. Monma, Bellcore, USA J. Nešetril, ˇ Charles University, Czech Republic W. R. Pulleyblank, IBM T. J. Watson Research Center, USA A. Recski, Technical University of Budapest, Hungary C. C. Ribeiro, Catholic University of Rio de Janeiro, Brazil H. Sachs, Technische Universität Ilmenau, Germany A. Schrijver, CWI, The Netherlands R. Shamir, Tel Aviv University, Israel N. J. A. Sloane, AT&T Research, USA W. T. Trotter, Arizona State University, USA D. J. A. Welsh, University of Oxford, United Kingdom D. de Werra, École Polytechnique Fédérale de Lausanne, Switzerland P. M. Winkler, Bell Labs, Lucent Technologies, USA Yue Minyi, Academia Sinica, People’s Republic of China

Series Volumes Dömösi, P. and Nehaniv, C. L., Algebraic Theory of Automata Networks Murota, K., Discrete Convex Analysis Toth, P. and Vigo, D., The Vehicle Routing Problem Anthony, M., Discrete Mathematics of Neural Networks: Selected Topics Creignou, N., Khanna, S., and Sudan, M., Complexity Classifications of Boolean Constraint Satisfaction Problems Hubert, L., Arabie, P., and Meulman, J., Combinatorial Data Analysis: Optimization by Dynamic Programming Peleg, D., Distributed Computing: A Locality-Sensitive Approach Wegener, I., Branching Programs and Binary Decision Diagrams: Theory and Applications Brandstädt, A., Le, V. B., and Spinrad, J. P., Graph Classes: A Survey McKee, T. A. and McMorris, F. R., Topics in Intersection Graph Theory Grilli di Cortona, P., Manzi, C., Pennisi, A., Ricca, F., and Simeone, B., Evaluation and Optimization of Electoral Systems

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Discrete Convex Analysis



Kazuo Murota



University of Tokyo Tokyo, Japan

Society for Industrial and Applied Mathematics Philadelphia

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Copyright © 2003 by the Society for Industrial and Applied Mathematics. 10 9 8 7 6 5 4 3 2 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Library of Congress Cataloging-in-Publication Data Murota, Kazuo, 1955Discrete convex analysis / Kazuo Murota. p. cm. — (SIAM monographs on discrete mathematics and applications) Includes bibliographical references and index. ISBN 978-1-611972-55-9 1. Convex functions. 2. Convex sets. 3. Mathematical analysis. I. Title. II. Series. QA331.5.M87 2003 515’.8—dc21 2003042468

is a registered trademark.

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Contents List of Figures

xi

Notation

xiii

Preface

xxi

1

2

Introduction to the Central Concepts 1.1 Aim and History of Discrete Convex Analysis . . 1.1.1 Aim . . . . . . . . . . . . . . . . . . . 1.1.2 History . . . . . . . . . . . . . . . . . 1.2 Useful Properties of Convex Functions . . . . . . 1.3 Submodular Functions and Base Polyhedra . . . . 1.3.1 Submodular Functions . . . . . . . . . 1.3.2 Base Polyhedra . . . . . . . . . . . . . 1.4 Discrete Convex Functions . . . . . . . . . . . . . 1.4.1 L-Convex Functions . . . . . . . . . . 1.4.2 M-Convex Functions . . . . . . . . . . 1.4.3 Conjugacy . . . . . . . . . . . . . . . 1.4.4 Duality . . . . . . . . . . . . . . . . . 1.4.5 Classes of Discrete Convex Functions Bibliographical Notes . . . . . . . . . . . . . . . . . . . .

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1 1 1 5 9 15 16 18 21 21 25 29 32 36 36

Convex Functions with Combinatorial Structures 2.1 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Convex Quadratic Functions . . . . . . . . . . . 2.1.2 Symmetric M-Matrices . . . . . . . . . . . . . . 2.1.3 Combinatorial Property of Conjugate Functions 2.1.4 General Quadratic L-/M-Convex Functions . . . 2.2 Nonlinear Networks . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Real-Valued Flows . . . . . . . . . . . . . . . . . 2.2.2 Integer-Valued Flows . . . . . . . . . . . . . . . 2.2.3 Technical Supplements . . . . . . . . . . . . . . . 2.3 Substitutes and Complements in Network Flows . . . . . . . 2.3.1 Convexity and Submodularity . . . . . . . . . .

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39 39 39 41 47 51 52 52 56 58 61 61

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vi

Contents 2.3.2 Technical Supplements . . . . Matroids . . . . . . . . . . . . . . . . . . 2.4.1 From Matrices to Matroids . 2.4.2 From Polynomial Matrices to Bibliographical Notes . . . . . . . . . . . . . . . 2.4

3

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Convex Analysis, Linear Programming, and Integrality 3.1 Convex Analysis . . . . . . . . . . . . . . . . . . . . . . 3.2 Linear Programming . . . . . . . . . . . . . . . . . . . 3.3 Integrality for a Pair of Integral Polyhedra . . . . . . . 3.4 Integrally Convex Functions . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . .

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63 68 68 71 74

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77 77 86 90 92 99

M-Convex Sets and Submodular Set Functions 4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Exchange Axioms . . . . . . . . . . . . . . . . . . . . 4.3 Submodular Functions and Base Polyhedra . . . . . . 4.4 Polyhedral Description of M-Convex Sets . . . . . . . 4.5 Submodular Functions as Discrete Convex Functions 4.6 M-Convex Sets as Discrete Convex Sets . . . . . . . . 4.7 M -Convex Sets . . . . . . . . . . . . . . . . . . . . . 4.8 M-Convex Polyhedra . . . . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . .

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101 101 102 103 108 111 114 116 118 119

L-Convex Sets and Distance Functions 5.1 Definition . . . . . . . . . . . . . . . . . . . . 5.2 Distance Functions and Associated Polyhedra 5.3 Polyhedral Description of L-Convex Sets . . . 5.4 L-Convex Sets as Discrete Convex Sets . . . . 5.5 L -Convex Sets . . . . . . . . . . . . . . . . . 5.6 L-Convex Polyhedra . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . .

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121 121 122 123 125 128 131 131

M-Convex Functions 6.1 M-Convex Functions and M -Convex Functions 6.2 Local Exchange Axiom . . . . . . . . . . . . . . 6.3 Examples . . . . . . . . . . . . . . . . . . . . . . 6.4 Basic Operations . . . . . . . . . . . . . . . . . 6.5 Supermodularity . . . . . . . . . . . . . . . . . . 6.6 Descent Directions . . . . . . . . . . . . . . . . . 6.7 Minimizers . . . . . . . . . . . . . . . . . . . . . 6.8 Gross Substitutes Property . . . . . . . . . . . . 6.9 Proximity Theorem . . . . . . . . . . . . . . . . 6.10 Convex Extension . . . . . . . . . . . . . . . . . 6.11 Polyhedral M-Convex Functions . . . . . . . . . 6.12 Positively Homogeneous M-Convex Functions .

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133 133 135 138 142 145 146 148 152 156 158 160 164

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6.13 Directional Derivatives and Subgradients . . . . . . . . . . . . . 166 6.14 Quasi M-Convex Functions . . . . . . . . . . . . . . . . . . . . . 168 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7

8

9

L-Convex Functions 7.1 L-Convex Functions and L -Convex Functions 7.2 Discrete Midpoint Convexity . . . . . . . . . . 7.3 Examples . . . . . . . . . . . . . . . . . . . . . 7.4 Basic Operations . . . . . . . . . . . . . . . . 7.5 Minimizers . . . . . . . . . . . . . . . . . . . . 7.6 Proximity Theorem . . . . . . . . . . . . . . . 7.7 Convex Extension . . . . . . . . . . . . . . . . 7.8 Polyhedral L-Convex Functions . . . . . . . . 7.9 Positively Homogeneous L-Convex Functions . 7.10 Directional Derivatives and Subgradients . . . 7.11 Quasi L-Convex Functions . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . .

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177 177 180 181 183 185 186 187 189 193 196 198 202

Conjugacy and Duality 8.1 Conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Submodularity under Conjugacy . . . . . . . 8.1.2 Polyhedral M-/L-Convex Functions . . . . . 8.1.3 Integral M-/L-Convex Functions . . . . . . . 8.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Separation Theorems . . . . . . . . . . . . . 8.2.2 Fenchel-Type Duality Theorem . . . . . . . . 8.2.3 Implications . . . . . . . . . . . . . . . . . . 8.3 M2 -Convex Functions and L2 -Convex Functions . . . . . 8.3.1 M2 -Convex Functions . . . . . . . . . . . . . 8.3.2 L2 -Convex Functions . . . . . . . . . . . . . . 8.3.3 Relationship . . . . . . . . . . . . . . . . . . 8.4 Lagrange Duality for Optimization . . . . . . . . . . . . 8.4.1 Outline . . . . . . . . . . . . . . . . . . . . . 8.4.2 General Duality Framework . . . . . . . . . . 8.4.3 Lagrangian Function Based on M-Convexity 8.4.4 Symmetry in Duality . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . .

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205 205 206 208 212 216 216 221 224 226 226 229 234 234 234 235 238 241 244

Network Flows 9.1 Minimum Cost Flow and Fenchel Duality . . 9.1.1 Minimum Cost Flow Problem . . 9.1.2 Feasibility . . . . . . . . . . . . . 9.1.3 Optimality Criteria . . . . . . . 9.1.4 Relationship to Fenchel Duality . 9.2 M-Convex Submodular Flow Problem . . . . 9.3 Feasibility of Submodular Flow Problem . .

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245 245 245 247 248 253 255 258

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Contents 9.4 9.5

Optimality Criterion by Potentials . . . . Optimality Criterion by Negative Cycles 9.5.1 Negative-Cycle Criterion . . 9.5.2 Cycle Cancellation . . . . . . 9.6 Network Duality . . . . . . . . . . . . . . 9.6.1 Transformation by Networks 9.6.2 Technical Supplements . . . . Bibliographical Notes . . . . . . . . . . . . . . . 10

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260 263 263 265 268 269 273 278

Algorithms 10.1 Minimization of M-Convex Functions . . . . . . . . . . . . . . 10.1.1 Steepest Descent Algorithm . . . . . . . . . . . . . 10.1.2 Steepest Descent Scaling Algorithm . . . . . . . . 10.1.3 Domain Reduction Algorithm . . . . . . . . . . . . 10.1.4 Domain Reduction Scaling Algorithm . . . . . . . 10.2 Minimization of Submodular Set Functions . . . . . . . . . . . 10.2.1 Basic Framework . . . . . . . . . . . . . . . . . . . 10.2.2 Schrijver’s Algorithm . . . . . . . . . . . . . . . . 10.2.3 Iwata–Fleischer–Fujishige’s Algorithm . . . . . . . 10.3 Minimization of L-Convex Functions . . . . . . . . . . . . . . 10.3.1 Steepest Descent Algorithm . . . . . . . . . . . . . 10.3.2 Steepest Descent Scaling Algorithm . . . . . . . . 10.3.3 Reduction to Submodular Function Minimization . 10.4 Algorithms for M-Convex Submodular Flows . . . . . . . . . . 10.4.1 Two-Stage Algorithm . . . . . . . . . . . . . . . . 10.4.2 Successive Shortest Path Algorithm . . . . . . . . 10.4.3 Cycle-Canceling Algorithm . . . . . . . . . . . . . 10.4.4 Primal-Dual Algorithm . . . . . . . . . . . . . . . 10.4.5 Conjugate Scaling Algorithm . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . .

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281 281 281 283 284 286 288 288 293 296 305 305 308 308 308 309 311 312 313 318 321

Application to Mathematical Economics 11.1 Economic Model with Indivisible Commodities 11.2 Difficulty with Indivisibility . . . . . . . . . . 11.3 M -Concave Utility Functions . . . . . . . . . 11.4 Existence of Equilibria . . . . . . . . . . . . . 11.4.1 General Case . . . . . . . . . . . . 11.4.2 M -Convex Case . . . . . . . . . . 11.5 Computation of Equilibria . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . .

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323 323 327 330 334 334 337 340 344

Application to Systems Analysis by Mixed Matrices 12.1 Two Kinds of Numbers . . . . . . . . . . . . . . . . . 12.2 Mixed Matrices and Mixed Polynomial Matrices . . . 12.3 Rank of Mixed Matrices . . . . . . . . . . . . . . . . 12.4 Degree of Determinant of Mixed Polynomial Matrices

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347 347 353 356 359

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ix

Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 Bibliography

363

Index

381

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List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15

Convex set and nonconvex set . . . . . . . . . . . . . . . . . . . . Convex function . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conjugate function (Legendre–Fenchel transform) . . . . . . . . . Separation for convex and concave functions . . . . . . . . . . . . Discrete separation . . . . . . . . . . . . . . . . . . . . . . . . . . . Convex and nonconvex discrete functions . . . . . . . . . . . . . . Exchange property (B-EXC[Z]) . . . . . . . . . . . . . . . . . . . . Definition of L-convexity . . . . . . . . . . . . . . . . . . . . . . . Discrete midpoint convexity . . . . . . . . . . . . . . . . . . . . . . Property of a convex function . . . . . . . . . . . . . . . . . . . . . Exchange property in the definition of M-convexity . . . . . . . . Conjugacy in discrete convexity . . . . . . . . . . . . . . . . . . . Duality theorems (f : M -convex function, h: M -concave function) Separation for convex sets . . . . . . . . . . . . . . . . . . . . . . . Classes of discrete convex functions (M -convex ∩ L -convex = M2 -convex ∩ L2 -convex = separable convex) . . . . . . . . . . . .

2 2 11 12 14 14 19 22 23 26 27 31 35 35

2.1 2.2 2.3 2.4 2.5

Electrical network . . . . . . . . . . . . . Multiterminal network . . . . . . . . . . . Characteristic curve . . . . . . . . . . . . Conjugate discrete convex functions fa (ξ) Discrete characteristic curve Γa . . . . . .

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42 53 54 57 57

3.1 3.2 3.3 3.4 3.5 3.6

Conjugate function (Legendre–Fenchel transform) . Separation for convex sets . . . . . . . . . . . . . . . Separation for convex and concave functions . . . . Nonconvexity in Minkowski sum . . . . . . . . . . . Integral neighborhood N (x) of x (◦: point of N (x)) Concept of integrally convex sets . . . . . . . . . . .

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81 83 84 91 94 97

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M -convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.1 5.2

L -convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Discrete midpoint convexity . . . . . . . . . . . . . . . . . . . . . . 129 xi

37

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List of Figures 6.1 6.2 6.3

Scaling f α for α = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Minimum spanning tree problem . . . . . . . . . . . . . . . . . . . 149 Quasi-convex function . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.1

Discrete midpoint convexity . . . . . . . . . . . . . . . . . . . . . . 181

8.1 8.2

Conjugacy in discrete convex functions . . . . . . . . . . . . . . . 215 Duality theorems (f : M -convex function, h: M -concave function) 224

9.1 9.2 9.3 9.4 9.5 9.6

Characteristic curve (kilter diagram) for linear cost . . . . . Minimum cost flow problem for Fenchel duality . . . . . . . . Submodular flow problem for M-convex intersection problem Transformation by a network . . . . . . . . . . . . . . . . . . Bipartite graphs for aggregation and convolution operations . Rooted directed tree for a laminar family . . . . . . . . . . .

10.1 10.2

ˆ at v  . . . . . . . . . . . . . . . . . . . . . . 316 Structure of G and G α Conjugate scaling f and scaling g α for α = 2 . . . . . . . . . . 320

11.1 11.2 11.3 11.4 11.5

Consumer’s behavior . . . . . . . . . . . . . . . . . . . . . . . . . . Exchange economy with no equilibrium for x◦ = (1, 1) . . . . . . . Minkowski sum D1 (p) + D2 (p) . . . . . . . . . . . . . . . . . . . . Aggregate cost function Ψ and its convex closure Ψ for an exchange economy with no equilibrium . . . . . . . . . . . . . . . . . . . . . Graph for computing a competitive equilibrium . . . . . . . . . . .

336 341

12.1 12.2 12.3 12.4 12.5

Electrical network with mutual couplings . . . . . . Hypothetical ethylene dichloride production system Jacobian matrix in the chemical process simulation Mechanical system . . . . . . . . . . . . . . . . . . . Accurate numbers . . . . . . . . . . . . . . . . . . .

348 350 351 352 353

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251 254 265 269 272 273

325 328 329

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Notation M-convex L-convex

function

set

f ∈M g∈L

B = B(ρ) ∈ M0 D = D(γ) ∈ L0

positively homogeneous function γˆ ∈ 0 M ρˆ ∈ 0 L

combinatorial function γ∈T ρ∈S

0: = (0, 0, . . . , 0) 1: = (1, 1, . . . , 1) 2V : set of all subsets of set V (i.e., power set of V ) ∀: “for all,” “for any,” or “for each” ∃: “there exists” or “for some”  : transpose of a vector or a matrix +: sum, Minkowski sum (3.21), (3.52) (3.20) 2 : infimal convolution over Rn n 2Z : infimal convolution over Z (i.e., integer infimal convolution) (6.43) ∨: componentwise maximum (1.28) ∨: “join” operation in a lattice Note 10.15 ∧: componentwise minimum (1.28) ∧: “meet” operation in a lattice Note 10.15 | · |: cardinality (number of elements) of a set [·, ·]: interval (of reals or integers) (3.1), (3.54) [·, ·]R : interval of real numbers (3.1) [·, ·]Z : interval of integers (3.54) ·, · : inner product, pairing (1.7), (3.18) (4.2) || · ||1 : 1 -norm of a vector || · ||∞ : ∞ -norm of a vector (3.60) · : convex hull of a set, convex closure of a function (3.56) · : rounding up to the nearest integer section 3.4

· : rounding down to the nearest integer section 3.4 ∂R f (x): subdifferential of (convex) function f at x (3.23), (6.86) ∂Z f (x): integral subdifferential of (convex) function f at x (6.88)  h(x): subdifferential of (concave) function h at x (8.19) ∂R ∂Z h(x): integral subdifferential of (concave) function h at x (8.19) + ∂ a: initial vertex of arc a section 2.2 section 2.2 ∂ − a: terminal vertex of arc a ∂ξ: boundary of flow ξ (2.27) ∂Ξ: set of boundaries of feasible flows section 9.3 xiii

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xiv

Notation

section 9.4 ∂Ξ∗ : set of boundaries of optimal flows ξ: flow section 2.2, section 9.1 ξ: current section 2.2 η: tension section 2.2, section 9.1 η: voltage section 2.2 δ + v: set of arcs leaving vertex v section 2.2 − section 2.2 δ v: set of arcs entering vertex v δp: coboundary of potential p (2.28), (9.20) δS : indicator function of set S (3.12), (3.51) δS • : support function of set S (3.31) Δ: Laplacian section 2.1.2 (6.2) Δf (x; v, u): directional difference of f at x in the direction of χv − χu + Δ X: set of arcs leaving vertex subset X (9.14) (9.15) Δ− X: set of arcs entering vertex subset X γ: distance function section 5.2 γ: shortest path length with respect to distance function γ section 5.2 γˆ : extension of distance function γ (6.82) γ: cost per unit flow section 9.1.1 (9.33) γp : reduced cost (9.2), (9.38) Γ1 : cost function in network flow problem (9.42) Γ2 : cost function in network flow problem Γ3 : cost function in network flow problem (9.7), (9.46) (2.31) Γa : characteristic curve of arc a κ: cut function (9.16) μ: supermodular set function section 4.3 section 9.4 Π∗ : set of optimal potentials ρ: submodular set function (4.9) ρ: rank function of a matroid section 2.4 ρˆ: Lov´asz extension (linear extension) of set function ρ (4.6) χ0 : zero vector section 2.1.3 section 2.1.3 χi : ith unit vector (1.14) χX : characteristic vector of subset X ω: valuation of a matroid section 2.4.2 aff S: affine hull of set S arg max f : set of maximizers of function f arg min f : set of minimizers of function f A[J]: submatrix of matrix A with column indices in J

section 3.1

B: M-convex set, M-convex polyhedron (B): simultaneous exchange axiom of matroids

section 4.1 section 2.4

(3.16) section 2.4

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Notation

xv

B: base family of a matroid B(ρ): base polyhedron defined by submodular set function ρ (B-EXC[R]): exchange axiom of M-convex polyhedra (B-EXC+ [R]): exchange axiom of M-convex polyhedra (B-EXC[Z]): exchange axiom of M-convex sets (B-EXCw [Z]): exchange axiom of M-convex sets (B-EXC+ [Z]): exchange axiom of M-convex sets (B-EXC− [Z]): exchange axiom of M-convex sets (B -EXC[Z]): exchange axiom of M -convex sets

section 2.4 (4.13) section 4.8 section 4.8 section 4.1 section 4.2 section 4.2 section 4.2 section 4.7

c: upper capacity function section 9.2 c: lower capacity function section 9.2 section 11.1 Cl : cost function of producer l C[Z → R]: set of univariate discrete convex functions (3.68) C[Z → Z]: set of univariate integer-valued discrete convex functions section 3.4 C[R → R]: set of univariate polyhedral convex functions section 3.1 C[Z|R → R]: set of univariate integral polyhedral convex functions section 6.11 C[R → R|Z]: set of univariate dual-integral polyhedral convex functions section 6.11 D: L-convex set, L-convex polyhedron section 5.1, section 5.6 section 11.1 Dh : demand function of consumer h D(γ): L-convex polyhedron defined by distance function γ (5.4) D(x): family of tight sets for base x (4.22) deg: degree of a polynomial section 2.4.2 dep(x, u): smallest tight set for base x that contains element u Note 10.11 det: determinant of a matrix section 2.4.1 dom ρ: effective domain of set function ρ (4.3) dom f : effective domain of function f on Rn or Zn (3.3), (1.25), (1.26) (1.26) domR f : effective domain of function f on Rn domZ f : effective domain of function f on Zn (1.25) epi f : epigraph of function f f : convex function, M-convex function f  (x; ·): directional derivative of function f at x f : convex closure of function f f˜: local convex extension of function f fˇ(x, y): a lower bound for f (y) − f (x) f • : (convex) conjugate of function f f •• : biconjugate (f • )• of function f f[a,b] : restriction of function f to interval [a, b]

(3.14) section 3.1, section 1.4.2 (3.24) (3.56) (3.61) (6.55) (3.26), (8.11) section 3.1 (3.55)

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xvi

Notation

fU : restriction of function f to subset U f U : projection of function f to subset U f U∗ : aggregation of function f to subset U f α : scaling of function f f α : conjugate scaling of function f f [−p](x): = f (x) − p, x

F : a field F (s): field of rational functions in variable s over F

(6.40) (6.41) (6.42) (6.47) (10.77) (3.22), (3.69) section 12.2 section 12.2

g: L-convex function section 1.4.1 G = (V, A): directed graph with vertex set V and arc set A section 2.2, section 9.2 h: concave function, M-concave function h◦ : (concave) conjugate of function h H: set of consumers

section 3.1, section 8.2 (3.28), (8.12) section 11.1

inf: infimum k: L-concave function K: set of indivisible commodities K: subfield of field F K(s): field of rational functions in variable s over K

section 8.2 section 11.1 section 12.2 section 12.2

L: set of producers L0 [Z]: set of L-convex sets L˜0 [Z]: set of indicator functions of L-convex sets L0 [R]: set of L-convex polyhedra L0 [Z|R]: set of integral L-convex polyhedra

section 11.1 section 5.1 section 1.4.3 section 5.6 section 5.6

L0 [Z]: set of L -convex sets

section 5.5

L0 [R]: set of L -convex polyhedra

section 5.6

L0 [Z|R]: set of integral L -convex polyhedra L[Z → R]: set of L-convex functions L[Z → Z]: set of integer-valued L-convex functions L[R → R]: set of polyhedral L-convex functions L[Z|R → R]: set of integral polyhedral L-convex functions L[R → R|Z]: set of dual-integral polyhedral L-convex functions L [Z → R]: set of L -convex functions L [Z → Z]: set of integer-valued L -convex functions L [R → R]: set of polyhedral L -convex functions L [Z|R → R]: set of integral polyhedral L -convex functions L [R → R|Z]: set of dual-integral polyhedral L -convex functions

section 5.6 section 7.1 section 7.1 section 7.8 section 7.8 section 7.8 section 7.1 section 7.1 section 7.8 section 7.8 section 8.1.2

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Notation

xvii

0 L[R

→ R]: set of positively homogeneous polyhedral L-convex functions section 7.9 L[Z|R → R]: set of positively homogeneous integral polyhedral L-convex 0 functions section 7.9 0 L[R → R|Z]: set of positively homogeneous dual-integral polyhedral L-convex functions section 8.1.2 L[Z → R]: set of positively homogeneous L-convex functions section 7.9 0 0 L[Z → Z]: set of positively homogeneous integer-valued L-convex functions section 7.9 section 8.3 L2 [Z → R]: set of L2 -convex functions section 8.3 L2 [Z → Z]: set of integer-valued L2 -convex functions L2 [Z → R]: set of L2 -convex functions

section 8.3

L2 [Z 

→ Z]: set of integer-valued functions  (L -APR[Z]): property of L -convex functions

section 8.3 section 7.2

max: maximum min: minimum M0 [Z]: set of M-convex sets ˜ 0 [Z]: set of indicator functions of M-convex sets M M0 [R]: set of M-convex polyhedra M0 [Z|R]: set of integral M-convex polyhedra

section 4.1 (1.21) section 4.8 section 4.8

L2 -convex

M0 [Z]: set of M -convex sets

section 4.7

M0 [R]: set of M -convex polyhedra

section 4.8

section 4.8 M0 [Z|R]: set of integral M -convex polyhedra M[Z → R]: set of M-convex functions section 6.1 M[Z → Z]: set of integer-valued M-convex functions section 6.1 M[R → R]: set of polyhedral M-convex functions section 6.11 M[Z|R → R]: set of integral polyhedral M-convex functions section 6.11 M[R → R|Z]: set of dual-integral polyhedral M-convex functions section 6.11 section 6.1 M [Z → R]: set of M -convex functions section 6.1 M [Z → Z]: set of integer-valued M -convex functions   section 6.11 M [R → R]: set of polyhedral M -convex functions section 6.11 M [Z|R → R]: set of integral polyhedral M -convex functions   M [R → R|Z]: set of dual-integral polyhedral M -convex functions section 8.1.2 0 M[R → R]: set of positively homogeneous polyhedral M-convex functions section 6.12 0 M[Z|R → R]: set of positively homogeneous integral polyhedral M-convex functions section 6.12 0 M[R → R|Z]: set of positively homogeneous dual-integral polyhedral M-convex functions section 8.1.2 M[Z → R]: set of positively homogeneous M-convex functions section 6.12 0

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xviii

Notation

0 M[Z

→ Z]: set of positively homogeneous integer-valued M-convex functions section 6.12 section 8.3 M2 [Z → R]: set of M2 -convex functions section 8.3 M2 [Z → Z]: set of integer-valued M2 -convex functions M2 [Z → R]: set of M2 -convex functions M2 [Z

section 8.3

M2 -convex

→ Z]: set of integer-valued functions section 8.3 (M-EXC[Z]): exchange axiom of M-convex functions section 1.4.2, section 6.1 section 1.4.2, section 6.1 (M-EXC [Z]): exchange axiom of M-convex functions section 6.2 (M-EXCloc [Z]): local exchange axiom of M-convex functions (M-EXCw [Z]): weak exchange axiom of M-convex functions section 6.2 (M-EXC[R]): exchange axiom of polyhedral M-convex functions section 1.4.2, section 6.11  section 6.11 (M-EXC [R]): exchange axiom of polyhedral M-convex functions (M -EXC[Z]): exchange axiom of M -convex functions section 1.4.2, section 6.1 (M -EXC[R]): exchange axiom of polyhedral M -convex functions section 1.4.2, section 6.11 (M -EXC [R]): exchange axiom of polyhedral M -convex functions section 6.11 (M -EXC+ [R]): exchange axiom of polyhedral M -convex functions section 2.1.3 (M -EXCd[R]): exchange axiom of polyhedral M -convex functions section 2.1.3  (M -EXC+ d [R]): exchange axiom of polyhedral M -convex functions section 2.1.3 (M-GS[Z]): gross substitutes property of M-convex functions section 6.8   section 6.8 (M -GS[Z]): gross substitutes property of M -convex functions (M -SWGS[Z]): stepwise gross substitutes property of M -convex functions section 6.8 (M-SI[Z]): descent property of M-convex functions section 6.6 section 6.6 (M -SI[Z]): descent property of M -convex functions section 11.3 (−M -EXC[Z]): exchange axiom of M -concave functions section 11.3 (−M -GS[Z]): gross substitutes property of M -concave function   (−M -SWGS[Z]): stepwise gross substitutes property of M -concave functions section 11.3 section 11.3 (−M -SI[Z]): ascent property of M -concave functions section 9.1.1 MCFP0 : minimum cost flow problem (linear arc cost) section 9.1.1 MCFP3 : minimum cost flow problem (nonlinear cost) section 9.2 MSFP1 : submodular flow problem (linear arc cost) section 9.2 MSFP2 : M-convex submodular flow problem (linear arc cost) MSFP3 : M-convex submodular flow problem (nonlinear arc cost) section 9.2 maxSFP: maximum submodular flow problem section 9.3 N (x): integral neighborhood of point x p: variable of an L-convex function

(3.58) section 1.4.1

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Notation

xix

p: potential section 2.2, section 9.4 p ∨ q: vector of componentwise maxima of p and q (1.28) p ∧ q: vector of componentwise minima of p and q (1.28) P(ρ): submodular polyhedron defined by submodular set function ρ (4.28) (5.16) P(γ, γˆ , γˇ ): L -convex polyhedron defined by (γ, γˆ , γˇ ) Q: set of rational numbers Q(ρ, μ): M -convex polyhedron (g-polymatroid) defined by (ρ, μ)

(4.36)

R: set of real numbers R+ : set of nonnegative real numbers R++ : set of positive real numbers ri S: relative interior of set S sup: supremum supp+ : positive support supp− : negative support S: convex hull of set S S[R]: set of real-valued submodular set functions S[Z]: set of integer-valued submodular set functions section (SBF[Z]): submodularity of functions on Zn n section (SBF[R]): submodularity of functions on R (SBF [Z]): translation submodularity of functions on Zn section  n (SBF [R]): translation submodularity of functions on R section section (SBS[Z]): submodularity of sets in Zn n (SBS[R]): submodularity of sets in R (SBS [Z]): translation submodularity of sets in Zn (SBS [R]): translation submodularity of sets in Rn Sl : supply function of producer l (SI): single improvement property

section 3.1

(2.21) (2.21) section 3.1 (4.10) (4.11) 1.4.1, section 7.1 1.4.1, section 7.8 1.4.1, section 7.1 1.4.1, section 7.8 1.4.1, section 5.1 section 5.6 section 5.5 section 5.6 section 11.1 section 11.3

T [R]: set of real-valued distance functions with triangle inequality T [Z]: set of integer-valued distance functions with triangle inequality section 1.4.1, (TRF[Z]): linearity in direction 1 of functions on Zn section 1.4.1, (TRF[R]): linearity in direction 1 of functions on Rn (TRS[Z]): translation property in direction 1 of sets in Zn section 1.4.1, (TRS[R]): translation property in direction 1 of sets in Rn Uh : utility function of consumer h

section section section section

5.2 5.2 7.1 7.8

section 5.1 section 5.6

section 11.1

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xx

Notation

(VM): axiom of valuated matroids

section 2.4.2

x: variable of an M-convex function x◦ : total initial endowment

section 1.4.2 (11.13)

Z: set of integers Z+ : set of nonnegative integers Z++ : set of positive integers

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Preface Discrete Convex Analysis is aimed at establishing a novel theoretical framework for solvable discrete optimization problems by means of a combination of the ideas in continuous optimization and combinatorial optimization. The theoretical framework of convex analysis is adapted to discrete settings and the mathematical results in matroid/submodular function theory are generalized. Viewed from the continuous side, the theory can be classified as a theory of convex functions f : Rn → R that have additional combinatorial properties. Viewed from the discrete side, it is a theory of discrete functions f : Zn → Z that enjoy certain nice properties comparable to convexity. Symbolically, Discrete convex analysis = Convex analysis + Matroid theory. The theory emphasizes duality and conjugacy as well as algorithms. This results in a novel duality framework for nonlinear integer programming. Two convexity concepts, called L-convexity and M-convexity, play primary roles, where “L” stands for “lattice” and “M” for “matroid.” L-convex functions and M-convex functions are convex functions with additional combinatorial properties distinguished by “L” and “M,” which are conjugate to each other through a discrete version of the Legendre–Fenchel transformation. L-convex functions and M-convex functions generalize, respectively, the concepts of submodular set functions and base polyhedra of (poly)matroids. L-convexity and M-convexity prevail in discrete systems. • In network flow problems, flow and tension are dual objects. Roughly speaking, flow corresponds to M-convexity and tension to L-convexity. • In matroids, the rank function corresponds to L-convexity and the base family to M-convexity. • M-matrices in matrix theory correspond to L-convexity and their inverses to M-convexity. Hence, in a discretization of the Poisson problem of partial differential equations, for example, the differential operator corresponds to L-convexity and the Green function to M-convexity. • Dirichlet forms in probability theory are essentially the same as quadratic L-convex functions. This book is intended to be read profitably by graduate students in operations research, mathematics, and computer science and also by mathematics-oriented xxi

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xxii

Preface

practitioners and application-oriented mathematicians. Self-contained presentation is envisaged. In particular, no familiarity with matroid theory nor with convex analysis is assumed. On the contrary, I hope the reader will acquire a unified view on matroids and convex functions through a variety of examples of discrete systems and the axiomatic approach presented in this book. I would like to express my appreciation for the encouragement, support, help, and criticism that I have received during my research on the theory of Discrete Convex Analysis. Joint work with Akiyoshi Shioura and Akihisa Tamura has been most substantial and collaborations with Satoru Fujishige, Satoru Iwata, Gleb Koshevoy, and Satoko Moriguchi enjoyable. Moral support offered by Bill Cunningham, Andr´as Frank, and L´ aci Lov´ asz has been encouraging. I have benefited from discussions with and/or comments by Andreas Dress, Atsushi Kajii, Mamoru Kaneko, Takahiro Kawai, Takashi Kumagai, Tomomi Matsui, Makoto Matsumoto, Shiro Matuura, Tom McCormick, Yoichi Miyaoka, Kiyohito Nagano, Maurice Queyranne, Andr´ as Recski, Andr´ as Seb˝o, Maiko Shigeno, Masaaki Sugihara, Zoltan Szigeti, Takashi Takabatake, Yoichiro Takahashi, Tamaki Tanaka, Fabio Tardella, Levent Tun¸cel, Jens Vygen, Jun Wako, Walter Wenzel, Yoshitsugu Yamamoto, and Zaifu Yang. In preparing this book I have been supported by several friends. Among others, Akiyoshi Shioura and Akihisa Tamura went through the text and provided comments and Satoru Iwata agreed that his unpublished results be included in this book. A significant part of this book is based on my previous book [147] in Japanese published by Kyoritsu Publishing Company. Finally, I express my deep gratitude to Peter Hammer, the chief editor of this monograph series, for his support in the realization of this book. October 2002

Kazuo Murota

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Chapter 1

Introduction to the Central Concepts

Discrete Convex Analysis aims at establishing a new theoretical framework of discrete optimization through mathematical studies of convex functions with combinatorial structures or discrete functions with convexity structures. This chapter is a succinct introduction to the central issues discussed in this book, including the role of convexity in optimization, several classes of well-behaved discrete functions, and duality theorems. We start with an account of the aim and the history of discrete convex analysis.

1.1

Aim and History of Discrete Convex Analysis

The motive for Discrete Convex Analysis is explained in general terms of optimization. Also included in this section is a brief chronological account of discrete convex functions in relation to the theory of matroids and submodular functions.

1.1.1

Aim

An optimization problem, or a mathematical programming problem, may be expressed generically as follows: Minimize f (x) subject to x ∈ S. This means that we are to find an x that minimizes the value of f (x) subject to the constraint that x should belong to the set S. Both f and S are given as the problem data, whereas x is a variable to be determined. The function f is called the objective function and the set S the feasible set . In continuous optimization, the variable x typically denotes a finite-dimensional real vector, say, x ∈ Rn , and accordingly we have S ⊆ Rn and f : Rn → R (or f : S → R).1 An optimization problem with S a convex set and f a convex function 1 The

notation R means the set of all real numbers and Rn the set of n-dimensional real vectors.

1

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2

Chapter 1. Introduction to the Central Concepts

Figure 1.1. Convex set and nonconvex set .

Y

6

Y = f (x)

x

z

y



λx + (1 − λ)y Figure 1.2. Convex function.

is referred to as a convex program, where a set S is convex if the line segment joining any two points in S is contained in S (see Fig. 1.1) and a function f : S → R defined on a convex set S is convex if λf (x) + (1 − λ)f (y) ≥ f (λx + (1 − λ)y)

(1.1)

whenever x, y ∈ S and 0 ≤ λ ≤ 1 (see Fig. 1.2). Convex programs constitute a class of optimization problems that are tractable both theoretically and practically, with a firm theoretical basis provided by “convex analysis.” The tractability of convex programs is largely based on the following properties of convex functions: 1. Local optimality (or minimality) guarantees global optimality. This implies, in particular, that a global optimum can be found by descent algorithms. 2. Duality, such as the min-max relation or the separation theorem, holds good. This leads, for instance, to primal-dual algorithms using dual variables and also to sensitivity analysis in terms of dual variables.

sidca00si 2013/2/12 page 3

1.1. Aim and History of Discrete Convex Analysis

3

Some more details on these issues will be discussed in section 1.2. In discrete optimization (or combinatorial optimization), on the other hand, the variable x takes discrete values; most typically, x is an integer vector or a {0, 1}vector. Whereas almost all discrete optimization problems arising from practical applications are difficult to solve efficiently, network flow problems are recognized as tractable discrete optimization problems. In the minimum cost flow problem with linear arc costs, for instance, we have the following fundamental facts that render the problem tractable: 1. A flow is optimal if and only if it cannot be improved by augmentation along a cycle. This statement means that the global optimality of a solution can be characterized by the local optimality with respect to augmentation along a cycle. 2. A flow is optimal if and only if there exists a potential on the vertex set such that the reduced arc cost with respect to the potential is nonnegative on every arc. This is a duality statement characterizing the optimality of a flow in terms of the dual variable (potential). This provides the basis for primal-dual algorithms. In more abstract terms, it is accepted that the tractability of the network flow problems stems from the matroidal structure (or submodularity) inherent therein. Whereas the meaning of this statement will be substantiated later, it is mentioned at this point that a matroid is an abstract combinatorial object defined as a pair of a finite set, say, V , and a family B of subsets of V that satisfies certain abstract axioms. We refer to V as the ground set, a member of B as a base, and a subset of a base as an independent set. The matroid is considered to be fundamental in combinatorial optimization, which is evidenced by the following facts:2 1. A base is optimal with respect to a given weight vector if and only if it cannot be improved by an elementary exchange, which means a modification of a base B to another base (B \ {u}) ∪ {v} with u in B and v not in B. Thus, the local optimality with respect to elementary exchanges guarantees the global optimality. Moreover, an optimal base can be found by the so-called greedy algorithm, which may be compared to the steepest descent algorithm in nonlinear optimization. 2. Given a pair of matroids on a common ground set, the intersection problem is to find a common independent set of maximum cardinality. Edmonds’s intersection theorem is a min-max duality theorem that characterizes the maximum cardinality as the minimum of a submodular function defined by the rank functions of the matroids. With the above facts it is natural to think of matroidal structure as a discrete or combinatorial analogue of convexity. The connection of matroidal structure to convexity was formulated in the early 1980s as a relationship between submodular 2A

more specific account of these facts will be given in section 1.3.

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4

Chapter 1. Introduction to the Central Concepts

functions and convex functions. It was shown by Frank that Edmonds’s intersection theorem can be rewritten as a separation theorem for a pair of submodular/supermodular functions, with an integrality (discreteness) assertion for the separating hyperplane in the case of integer-valued functions. Another reformulation of Edmonds’s intersection theorem is Fujishige’s Fenchel-type min-max duality theorem for a pair of submodular/supermodular functions, again with an integrality assertion in the case of integer-valued functions. A precise statement, beyond analogy, about the relationship between submodular functions and convex functions was made by Lov´ asz: A set function is submodular if and only if the so-called Lov´ asz extension of that set function is convex. These results led to the recognition that the essence of the duality for submodular/supermodular functions consists of the discreteness (integrality) assertion in addition to the duality for convex/concave functions. Namely, Duality for submodular functions = Convexity + Discreteness. Such developments notwithstanding, our understanding of convexity in discrete optimization seems to be only partial. In convex programming, a convex objective function is minimized over a convex feasible region, which may be described by a system of inequalities in (other) convex functions. In matroid optimization, explained above, the objective function is restricted to be linear and the feasible region is described by a system of inequalities using submodular functions. This means that the convexity argument for submodular functions applies to the convexity of feasible regions and not to the convexity of objective functions. In the literature, however, we can find a number of nice structural results on discrete optimization of nonlinear objective functions. For example, the minimum cost flow problem with a separable convex cost function admits optimality criteria similar to those for linear arc costs (Minoux [131] and others), which can be carried over to the submodular flow problem with a separable convex cost function (Fujishige [65]). The minimization of a separable convex function over a base polyhedron also admits a local optimality criterion with respect to elementary exchanges (Fujishige [60], Girlich–Kowaljow [78], Groenevelt [81]). This fact is used in the literature of resource allocation problems (Ibaraki–Katoh [93], Hochbaum [90], Hochbaum–Hong [91], Girlich–Kovalev–Zaporozhets [77]). The convexity argument concerning submodular functions, however, does not help us understand these results in relation to convex analysis. We are thus waiting for a more general theoretical framework for discrete optimization that can be compared to convex analysis for continuous optimization. Discrete Convex Analysis is aimed at establishing a general theoretical framework for solvable discrete optimization problems by means of a combination of the ideas in continuous optimization and combinatorial optimization. The theoretical framework of convex analysis is adapted to discrete settings and the mathematical results in matroid/submodular function theory are generalized. Viewed from the continuous side, the theory can be classified as a theory of convex functions f : Rn → R that have additional combinatorial properties. Viewed from the discrete side, it is a theory of discrete functions f : Zn → Z that enjoy certain nice

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1.1. Aim and History of Discrete Convex Analysis

5

properties comparable to convexity.3 Symbolically, Discrete convex analysis = Convex analysis + Matroid theory. The theory emphasizes duality and conjugacy with a view to providing a novel duality framework for nonlinear integer programming. It may be in order to mention that the present theory extends the direction set forth by J. Edmonds, A. Frank, S. Fujishige, and L. Lov´ asz (see section 1.1.2), but it is, rather, independent of the convexity arguments in the theories of greedoids, antimatroids, convex geometries, and oriented matroids (Bj¨orner–Las Vergnas–Sturmfels–White–Ziegler [16], Korte– Lov´ asz–Schrader [114]). Two convexity concepts, called L-convexity and M-convexity, play primary roles in the present theory. L-convex functions and M-convex functions are both (extensible to) convex functions and they are conjugate to each other through a discrete version of the Legendre–Fenchel transformation. L-convex functions and Mconvex functions generalize, respectively, the concepts of submodular set functions and base polyhedra. It is noted that the “L” in “L-convexity” stands for “lattice” and the “M” in “M-convexity” for “matroid.”

1.1.2

History

This section is devoted to an account of the history of discrete convex functions in matroid theory that led to L-convex and M-convex functions (see Table 1.1). There are, however, many other previous and recent studies on discrete convexity outside the literature of the matroid (Hochbaum–Shamir–Shanthikumar [92], Ibaraki–Katoh [93], Kindler [112], Miller [130], and so on). The concept of matroids was introduced by H. Whitney [218] in 1935, together with the equivalence between the submodularity of rank functions and the exchange property of independent sets. This equivalence is the germ of the conjugacy between L-convex and M-convex functions in the present theory of discrete convex analysis. In the late 1960s, J. Edmonds found a fundamental duality theorem on the intersection problem for a pair of (poly)matroids. This theorem, Edmonds’s intersection theorem, shows a min-max relation between the maximum of a common independent set and the minimum of a submodular function derived from the rank functions. The famous article of Edmonds [44] convinced us of the fundamental role of submodularity in discrete optimization. Analogies of submodular functions to convex functions and to concave functions were discussed at the same time. The min-max relation supported the analogy to convex functions, whereas some other facts pointed to concave functions. No unanimous conclusion was reached at this point. The relationship between submodular functions and convex functions, which was made clear in the early 1980s through the work of A. Frank, S. Fujishige, and L. Lov´ asz, was described in section 1.1.1 but is mentioned again in view of its importance. The fundamental relationship between submodular functions and convex functions, due to Lov´ asz [123], says that a set function is submodular if 3 The

notation Z means the set of all integers and Zn the set of n-dimensional integer vectors.

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Chapter 1. Introduction to the Central Concepts

Table 1.1. History (matroid and convexity). Year (ca.) 1935

Author(s) Whitney [218]

1965

Edmonds [44]

1975 Edmonds [45] Lawler [118] Tomizawa–Iri [201] Iri–Tomizawa [96] Frank [54] 1982

1990

1995

2000

Frank [55] Fujishige [62] Lov´ asz [123] Dress–Wenzel [41], [42] Favati–Tardella [49] Murota [135], [139] Murota [137], [140]

Murota–Shioura [151] Fujishige–Murota [68] Murota–Shioura [152] Murota–Shioura [156], [157]

Result axioms of matroid exchange property ⇔ submodularity polymatroid polyhedral method intersection theorem weighted matroid intersection

potential potential weight splitting relationship to convexity discrete separation theorem Fenchel-type duality Lov´ asz (linear) extension valuated matroid axiom, greedy algorithm integrally convex function valuated matroid intersection L-/M-convex function Fenchel-type duality separation theorem M -convex function L -convex function polyhedral L-/M-convex function continuous L-/M-convex function

and only if the Lov´ asz extension of that function is convex. Reformulations of Edmonds’s intersection theorem into a separation theorem for a pair of submodular/supermodular functions by Frank [55] and a Fenchel-type min-max duality theorem by Fujishige [62] indicate its similarity to convex analysis. The discrete mathematical content of these theorems, which cannot be captured by the relationship of submodularity to convexity, lies in the integrality assertion for integervalued submodular/supermodular functions. Further analogy to convex analysis, such as subgradients, was conceived by Fujishige [63]. These developments in the 1980s led us to the understanding that (i) submodularity should be compared to convexity, not to concavity, and (ii) the essence of the duality for a pair of submodular/supermodular functions lies in the discreteness (integrality) assertion in

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1.1. Aim and History of Discrete Convex Analysis

7

addition to the duality for convex/concave functions: (i) submodular functions  convex functions, (ii) duality for submodular functions  convexity + discreteness. A remark is in order here, although it involves technical terminology from convex analysis. The Lov´ asz extension of a submodular set function is a convex function, but it is bound to be positively homogeneous (f (λx) = λf (x) for λ ≥ 0). As a matter of fact, it coincides with the support function of the base polyhedra associated with the submodular function. This suggests that the convexity arguments on submodularity deal with a restricted class of convex functions, namely, the class of support functions of convex sets. The relationship of submodular set functions to convex functions summarized in (i) and (ii) above is generalized to the full extent by the concept of L-convex functions in the present theory. Addressing the issue of local vs. global optimality for functions defined on integer lattice points, P. Favati and F. Tardella [49] came up with the concept of integrally convex functions in 1990. This concept successfully captures a fairly general class of functions on integer lattice points, for which a local optimality implies the global optimality. Moreover, the class of submodular integrally convex functions (i.e., integrally convex functions that are submodular on integer lattice points) was considered as a subclass of integrally convex functions. It turns out that this concept is equivalent to a variant of L-convex functions, called L -convex functions, in the present theory. We have so far seen major milestones on the road toward L-convex functions and now turn to M-convex functions. A weighted version of the matroid intersection problem was introduced by Edmonds [44]. The problem is to find a maximum-weight common independent set (or a common base) with respect to a given weight vector. Efficient algorithms for this problem were developed in the 1970s by Edmonds [45], Lawler [118], Tomizawa– Iri [201], and Iri–Tomizawa [96] on the basis of a nice optimality criterion in terms of dual variables. The optimality criterion of Frank [54] in terms of weight splitting can be thought of as a version of such an optimality criterion using dual variables. The weighted matroid intersection problem was generalized to the polymatroid intersection problem as well as to the submodular flow problem. It should be noted, however, that in all of these generalizations the weighting remained linear or separable convex. The concept of valuated matroids, introduced by Dress and Wenzel [41], [42] in 1990, provides a nice framework for nonlinear optimization on matroids. A valuation of a matroid is a nonlinear and nonseparable function of bases satisfying a certain exchange axiom. It was shown by Dress and Wenzel that a version of the greedy algorithm works to maximize a matroid valuation and this property in turn characterizes a matroid valuation. Not only the greedy algorithm but also the intersection problem extends to valuated matroids. The valuated matroid intersection problem, introduced by Murota [135], is to maximize the sum of two valuations. This generalizes the weighted matroid intersection problem, since linear weighting is a special case of matroid valuation. Optimality criteria, such as weight splitting,

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8

Chapter 1. Introduction to the Central Concepts

as well as algorithms for the weighted matroid intersection, are generalized to the valuated matroid intersection (Murota [136]). An analogy of matroid valuations to concave functions resulted in a Fenchel-type min-max duality theorem for matroid valuations (Murota [139]). This Fenchel-type duality is neither a generalization nor a special case of Fujishige’s Fenchel-type duality for submodular functions, but these two can be generalized into a single min-max equation, which is the Fenchel-type duality theorem in the present theory. A further analogy of valuated matroids to concave functions led to the concept of M-convex/concave functions in Murota [137], 1996. M-convexity is a concept of “convexity” for functions defined on integer lattice points in terms of an exchange axiom and affords a common generalization of valuated matroids and (integral) polymatroids. A valuated matroid can be identified with an M-concave function defined on {0, 1}-vectors. The base polyhedron of an integral polymatroid is a synonym for a {0, +∞}-valued M-convex function. The valuated matroid intersection problem and the polymatroid intersection problem are unified into the M-convex intersection problem. The Fenchel-type duality theorem for matroid valuations is generalized for M-convex functions and the submodular flow problem is generalized to the M-convex submodular flow problem (Murota [142]), which involves an M-convex function as a nonlinear cost. The nice optimality criterion using dual variables survives in this generalization. Thus, M-convex functions yield fruitful generalizations of many important optimization problems on matroids. The two independent lines of development, namely, the convexity argument for submodular functions in the early 1980s and that for valuated matroids and M-convex functions in the early 1990s, were merged into a unified framework of discrete convex analysis advocated by Murota [140] in 1998. The concept of Lconvex functions was introduced as a generalization of submodular set functions. L-convex functions form a conjugate class of M-convex functions with respect to the Legendre–Fenchel transformation. This completes the picture of conjugacy advanced by Whitney [218] in 1935 as the equivalence between the submodularity of the rank function of a matroid and the exchange property of independent sets of a matroid. The duality theorems carry over to L-convex and M-convex functions. In particular, the separation theorem for L-convex functions is a generalization of Frank’s separation theorem for submodular functions. Ramifications of the concepts of L- and M-convexity followed. M -convex functions,4 introduced by Murota–Shioura [151], are essentially equivalent to M-convex functions, but are sometimes more convenient. For example, a convex function in one variable, when considered only for integer values of the variable, is an M -convex function that is not M-convex. L -convex functions, due to Fujishige–Murota [68], are an equivalent variant of L-convex functions. It turns out that L -convex functions are exactly the same as the submodular integrally convex functions that had been introduced by Favati–Tardella [49] in their study of local vs. global optimality. The success of polyhedral methods in combinatorial optimization naturally suggests the possibility of polyhedral versions of L- and M-convex functions. This idea was worked out by Murota–Shioura [152] with the introduction of the concepts of 4 “M -convex”

should be read “M-natural-convex” and similarly for “L -convex.”

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9

L- and M-convexity for polyhedral functions (piecewise linear functions in real variables). These convexity concepts were defined also for quadratic functions (Murota– Shioura [155]) and for closed convex functions (Murota–Shioura [156], [157]). We conclude this section with a remark on a subtle point in the relationship between submodularity and convexity. From the discussion in the early 1980s we have agreed that submodularity should be compared to convexity. This statement is certainly true for set functions. When it comes to functions on integer points, however, we need to be careful. As a matter of fact, an M -concave function is submodular and concave extensible (Theorems 6.19 and 6.42), whereas an L -convex function is submodular and convex extensible (Theorem 7.20). This shows that submodularity and convexity are mutually independent properties for functions on integer points. It is undoubtedly true, however, that submodularity is essentially related to discrete convexity.

1.2

Useful Properties of Convex Functions

We have already mentioned that convex functions are tractable in optimization (or minimization) problems, which is mainly because of the following properties: 1. Local optimality (or minimality) guarantees global optimality. 2. Duality, e.g., the min-max relation or the separation theorem, holds good. The purpose of this section is to give more specific descriptions of these properties and to discuss their possible versions for discrete functions. Let us first recall the definition of a convex function. A function f : Rn → R ∪ {+∞} is said to be convex if λf (x) + (1 − λ)f (y) ≥ f (λx + (1 − λ)y)

(1.2)

for all x, y ∈ Rn and for all λ with 0 ≤ λ ≤ 1, where it is understood that the inequality is satisfied if f (x) or f (y) is equal to +∞. The inequality (1.2) implies that the set S = {x ∈ Rn | f (x) < +∞}, called the effective domain of f , is a convex set. Hence, the present definition of a convex function coincides with the one in (1.1) that makes explicit reference to the effective domain S. A special case of inequality (1.2) for λ = 1/2 yields the midpoint convexity   x+y f (x) + f (y) ≥f (1.3) (x, y ∈ Rn ) 2 2 and, conversely, this implies convexity provided f is continuous. We often assume (explicitly or implicitly) that f (x) < +∞ for some x ∈ Rn whenever we talk about a convex function f . A function h : Rn → R ∪ {−∞} is said to be concave if −h is convex. A point (or vector) x is said to be a global optimum of f if the inequality f (x) ≤ f (y)

(1.4)

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Chapter 1. Introduction to the Central Concepts

holds for every y and a local optimum if this inequality holds for every y in some neighborhood of x. Obviously, global optimality implies local optimality. The converse is not true in general, but it is true for convex functions. Theorem 1.1. For a convex function, global optimality (or minimality) is guaranteed by local optimality. Proof. Let x be a local optimum of a convex function f . Then we have f (z) ≥ f (x) for any z in some neighborhood U of x. For any y, z = λx + (1 − λ)y belongs to U for λ < 1 sufficiently close to 1 and it follows from (1.2) that λf (x) + (1 − λ)f (y) ≥ f (λx + (1 − λ)y) = f (z) ≥ f (x). This implies f (y) ≥ f (x). The above theorem is significant and useful in that it reduces the global property to a local one. Still it refers to an infinite number of points or directions around x for the local optimality. In considering discrete structures on top of convexity we may hope that a fixed and finite set of directions suffices to guarantee the local optimality. For example, in the simplest case of a separable convex function f (x) =

n 

fi (x(i)),

(1.5)

i=1

which is the sum of univariate convex functions5 fi (x(i)) in each component of x = (x(i) | i = 1, . . . , n), it suffices to check for local optimality in 2n directions: the positive and negative directions of the coordinate axes. Such a phenomenon of discreteness in direction, so to speak, is a reflection of the combinatorial structure of separable convex functions. Although the combinatorial structure of separable convex functions is too simple for further serious consideration, similar phenomena of discreteness in direction occur in nontrivial ways for L-convex or M-convex functions, as we will see in section 1.4. We now go on to the second issue of duality and conjugacy. For a function f (not necessarily convex), the convex conjugate f • : Rn → R ∪ {+∞} is defined by f • (p) = sup{ p, x − f (x) | x ∈ Rn } where p, x =

n 

p(i)x(i)

(p ∈ Rn ),

(1.6)

(1.7)

i=1

for p = (p(i) | i = 1, . . . , n) and x = (x(i) | i = 1, . . . , n). The function f • is also referred to as the (convex) Legendre–Fenchel transform of f and the mapping f → f • as the (convex) Legendre–Fenchel transformation. 5A

univariate function means a function in a single variable.

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1.2. Useful Properties of Convex Functions

Y

6

11

Y = f (x)

−f • (p) Y = p, x − f • (p) x Figure 1.3. Conjugate function (Legendre–Fenchel transform).

For example, for f (x) = exp(x), where n = 1, we see ⎧ ⎨ p log p − p f • (p) = 0 ⎩ +∞

(p > 0), (p = 0), (p < 0)

by a simple calculation. See Fig. 1.3 for the geometric meaning in the case of n = 1. The Legendre–Fenchel transformation gives a one-to-one correspondence in the class of well-behaved convex functions, called closed proper convex functions, where the precise meaning of this technical terminology (not important here) will be explained later in section 3.1. The notation f •• means (f • )• , the conjugate of the conjugate function of f . Theorem 1.2 (Conjugacy). The Legendre–Fenchel transformation f → f • gives a symmetric one-to-one correspondence in the class of all closed proper convex functions. That is, for a closed proper convex function f , f • is a closed proper convex function and f •• = f . Similarly, for a function h, the concave conjugate h◦ : Rn → R ∪ {−∞} is defined by h◦ (p) = inf{ p, x − h(x) | x ∈ Rn }

(p ∈ Rn ).

(1.8)

The duality principle in convex analysis can be expressed in a number of different forms. One of the most appealing statements is in the form of the separation theorem, which asserts the existence of a separating affine function Y = α∗ + p∗ , x

for a pair of convex and concave functions (see Fig. 1.4). Theorem 1.3 (Separation theorem). Let f : Rn → R ∪ {+∞} and h : Rn → R∪{−∞} be convex and concave functions, respectively (satisfying certain regularity

sidca00si 2013/2/12 page 12

12

Chapter 1. Introduction to the Central Concepts Y = f (x)

Y = α∗ + p∗ , x

Y 6 Y = h(x) x Figure 1.4. Separation for convex and concave functions.

conditions). If 6 f (x) ≥ h(x)

(∀ x ∈ Rn ),

there exist α∗ ∈ R and p∗ ∈ Rn such that f (x) ≥ α∗ + p∗ , x ≥ h(x)

(∀ x ∈ Rn ).

It is admitted that the statement above is mathematically incomplete, referring to certain regularity conditions, which will be specified later in section 3.1. Another expression of the duality principle is in the form of the Fenchel duality. This is a min-max relation between a pair of convex and concave functions and their conjugate functions. The certain regularity conditions in the statement below will be specified later. Theorem 1.4 (Fenchel duality). Let f : Rn → R ∪ {+∞} and h : Rn → R ∪ {−∞} be convex and concave functions, respectively (satisfying certain regularity conditions). Then min{f (x) − h(x) | x ∈ Rn } = max{h◦ (p) − f • (p) | p ∈ Rn }. Such a min-max theorem is computationally useful in that it affords a certificate of optimality. Suppose that we want to minimize f (x) − h(x) and have x = x ˆ as a candidate for the minimizer. How can we verify or prove that x ˆ is indeed an optimal solution? One possible way is to find a vector pˆ such that p) − f • (ˆ p). This implies the optimality of x ˆ by virtue of the f (ˆ x) − h(ˆ x) = h◦ (ˆ min-max theorem. The vector pˆ, often called a dual optimal solution, serves as 6 The

notation ∀ means “for all,” “for any,” or “for each.”

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1.2. Useful Properties of Convex Functions

13

a certificate for the optimality of x ˆ. It is emphasized that the min-max theorem guarantees the existence of such a certificate pˆ for any optimal solution x ˆ. It is also mentioned that the min-max theorem does not tell us how to find optimal solutions x ˆ and pˆ. It is one of the recurrent themes in discrete convexity how the conjugacy and the duality above should be adapted in discrete settings. To be specific, let us consider integer-valued functions on integer lattice points and discuss possible notions of conjugacy and duality for f : Zn → Z ∪ {+∞} and h : Zn → Z ∪ {−∞}. Some ingredients of discreteness (integrality) are naturally expected in the formulation of conjugacy and duality. This amounts to discussing another kind of discreteness, discreteness in value, in contrast with discreteness in direction, mentioned above. Discrete versions of the Legendre–Fenchel transformations can be defined by f • (p) = sup{ p, x − f (x) | x ∈ Zn } ◦

h (p) = inf{ p, x − h(x) | x ∈ Z } n

(p ∈ Zn ), (p ∈ Z ). n

(1.9) (1.10)

They are meaningful as transformations of discrete functions in that the resulting functions f • and h◦ are also integer valued on integer points. We call (1.9) and (1.10), respectively, convex and concave discrete Legendre–Fenchel transformations. With these definitions, a discrete version of the Fenchel duality would read as follows. [Discrete Fenchel-type duality theorem] Let f : Zn → Z ∪ {+∞} and h : Zn → Z ∪ {−∞} be convex and concave functions, respectively (in an appropriate sense). Then min{f (x) − h(x) | x ∈ Zn } = max{h◦ (p) − f • (p) | p ∈ Zn }. Such a theorem, if any, claims a min-max duality relation for integer-valued nonlinear functions, which is not likely to be true for an arbitrary class of discrete functions. It is emphasized that the definition of convexity itself is left open in the above generic statement, although h should be called concave when −h is convex. As for the separation theorem, a possible discrete version would read as follows, imposing integrality (α∗ ∈ Z, p∗ ∈ Zn ) on the separating affine function (see Fig. 1.5). [Discrete separation theorem] Let f : Zn → Z∪{+∞} and h : Zn → Z∪ {−∞} be convex and concave functions, respectively (in an appropriate sense). If f (x) ≥ h(x) (∀ x ∈ Zn ), there exist α∗ ∈ Z and p∗ ∈ Zn such that f (x) ≥ α∗ + p∗ , x ≥ h(x)

(∀ x ∈ Zn ).

Again the precise definition of convexity remains unspecified here.

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Chapter 1. Introduction to the Central Concepts

Y = f (x)

Y = α∗ + p∗ , x

Y = h(x) Figure 1.5. Discrete separation. 6

6

f

f

x

x

Figure 1.6. Convex and nonconvex discrete functions.

To motivate the framework we will introduce in the subsequent sections, let us try a naive and natural candidate for the convexity concept, which turns out to be insufficient. Let us (temporarily) define f : Zn → Z ∪ {+∞} to be convex if it can be extended to a convex function on Rn , i.e., if there exists a convex function f : Rn → R ∪ {+∞} such that f (x) = f (x)

(x ∈ Zn ).

(1.11)

This is illustrated in Fig. 1.6. In the one-dimensional case (with n = 1), this is equivalent to defining f : Z → Z ∪ {+∞} to be convex if f (x − 1) + f (x + 1) ≥ 2f (x)

(∀ x ∈ Z).

(1.12)

As is easily verified, the discrete separation theorem, as well as the discrete Fenchel duality, holds with this definition in the case of n = 1. When it comes to higher dimensions, the situation is not that simple. The following examples demonstrate that the discrete separation fails with this naive definition of convexity.

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1.3. Submodular Functions and Base Polyhedra

15

Example 1.5 (Failure of discrete separation). Consider two discrete functions defined by f (x) = max(0, x(1) + x(2)),

h(x) = min(x(1), x(2)),

where x = (x(1), x(2)) ∈ Z2 . They are integer valued on the integer lattice Z2 , with f (0) = h(0) = 0, and can be extended, respectively, to a convex function f : R2 → R and a concave function h : R2 → R given by f (x) = max(0, x(1) + x(2)),

h(x) = min(x(1), x(2)),

where x = (x(1), x(2)) ∈ R2 . Since f (x) ≥ h(x) (∀ x ∈ R2 ), the separation theorem in convex analysis (Theorem 1.3) applies to the pair (f , h) to yield a (unique) separating affine function p∗ , x , with p∗ = (1/2, 1/2). We have f (x) ≥ p∗ , x ≥ h(x) for all x ∈ R2 and, a fortiori, f (x) ≥ p∗ , x ≥ h(x) for all x ∈ Z2 . However, there exists no integral vector p∗ ∈ Z2 such that f (x) ≥ p∗ , x ≥ h(x) for all x ∈ Z2 . This demonstrates the failure of the desired discreteness in the separating affine function. Example 1.6 (Failure of real-valued separation). This example shows that even the existence of a separating affine function can be denied. For the discrete functions f (x) = |x(1) + x(2) − 1|,

h(x) = 1 − |x(1) − x(2)|,

where x = (x(1), x(2)) ∈ Z2 , we have f (x) ≥ h(x) (∀ x ∈ Z2 ). There exists, however, no pair of real number α∗ ∈ R and real vector p∗ ∈ R2 for which f (x) ≥ α∗ + p∗ , x ≥ h(x) for all x ∈ Z2 . Note that the separation theorem in convex analysis (Theorem 1.3) does not apply to the pair of their convex/concave extensions (f , h), which are given by f (x) = |x(1) + x(2) − 1|,

h(x) = 1 − |x(1) − x(2)|

for x = (x(1), x(2)) ∈ R , since f (1/2, 1/2) < h(1/2, 1/2). This example also shows that f ≥ h on Rn does not follow from f ≥ h on Zn . 2

Similarly, the discrete Fenchel duality fails under the naive definition of convexity. The above two examples serve to demonstrate this. Thus, the naive approach to discrete convexity does not work, and some deep combinatorial or discrete-mathematical considerations are needed. We are now motivated to look at some results in the area of matroids and submodular functions, which we hope will provide a clue for fruitful definitions of discrete convexity.

1.3

Submodular Functions and Base Polyhedra

We describe here a few results on submodular functions and base polyhedra that are relevant to our discussion in this introductory chapter, whereas a more comprehensive treatment is given in section 4.3. Emphasis is placed on the conjugacy relationship between these two objects and the analogy to convex functions recognized in the early 1980s.

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1.3.1

Chapter 1. Introduction to the Central Concepts

Submodular Functions

A set function7 ρ : 2V → R ∪ {+∞}, which assigns a real number (or +∞) to each subset of a given finite set V , is said to be submodular if ρ(X) + ρ(Y ) ≥ ρ(X ∪ Y ) + ρ(X ∩ Y )

(∀ X, Y ⊆ V ),

(1.13)

where it is understood that the inequality is satisfied if ρ(X) or ρ(Y ) is equal to +∞. This is called the submodularity inequality. We assume, for a set function ρ in general, that ρ(∅) = 0 and ρ(V ) is finite. A function μ : 2V → R ∪ {−∞} is supermodular if −μ is submodular. The relationship between submodularity and convexity can be formulated in terms of the Lov´ asz extension (also called the Choquet integral or the linear extenasz extension of ρ is a sion). For any set function ρ : 2V → R ∪ {±∞} the Lov´ function ρˆ : RV → R ∪ {±∞}, a real-valued function in real variables, defined as follows.8 For each p ∈ RV , we index the elements of V in nonincreasing order in the components of p; i.e., V = {v1 , v2 , . . . , vn } and p(v1 ) ≥ p(v2 ) ≥ · · · ≥ p(vn ), where9 n = |V |. Using the notation pj = p(vj ), Vj = {v1 , v2 , . . . , vj } for j = 1, . . . , n, and χX for the characteristic vector of a subset X ⊆ V defined by  1 (v ∈ X), (1.14) χX (v) = 0 (v ∈ V \ X), we have p=

n−1 

(pj − pj+1 )χVj + pn χVn .

(1.15)

j=1

This is an expression of p as a linear combination of the characteristic vectors of the subsets Vj . The linear interpolation of ρ according to this expression yields ρˆ(p) =

n−1 

(pj − pj+1 )ρ(Vj ) + pn ρ(Vn ),

(1.16)

j=1

which is the definition of the Lov´ asz extension ρˆ of ρ. Note that 0 × (±∞) = 0 in (1.16) by convention. The Lov´asz extension ρˆ is indeed an extension of ρ in that ρˆ(χX ) = ρ(X) for X ⊆ V . The relationship between submodularity and convexity reads as follows.10 7 The notation 2V means the set of all subsets of V or the power set of V . Hence, X ∈ 2V is equivalent to saying that X is a subset of V . 8 The notation RV means the real vector space with coordinates indexed by the elements of V . If V consists of n elements, then RV may be identified with Rn . In the original definition, ρˆ(p) is defined only for nonnegative vectors p. 9 The notation |V | means the number of elements of V . 10 The proofs of Theorems 1.7 and 1.8 are given in Chapter 4, when we come to their rigorous treatments in Theorems 4.16 and 4.17.

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17

Theorem 1.7 (Lov´asz). A set function ρ is submodular if and only if its Lov´ asz extension ρˆ is convex. Duality for a pair of submodular/supermodular functions is formulated in the following discrete separation theorem. We use the notation  x(X) = x(v) v∈X

for a vector x = (x(v) | v ∈ V ) ∈ R

V

and a subset X ⊆ V .

Theorem 1.8 (Frank’s discrete separation theorem). Let ρ : 2V → R ∪ {+∞} and μ : 2V → R ∪ {−∞} be submodular and supermodular functions, respectively, with ρ(∅) = μ(∅) = 0, ρ(V ) < +∞, and μ(V ) > −∞. If ρ(X) ≥ μ(X) ∗

there exists x ∈ R

V

(∀ X ⊆ V ),

such that ρ(X) ≥ x∗ (X) ≥ μ(X)

(∀ X ⊆ V ).

(1.17)



Moreover, if ρ and μ are integer valued, the vector x can be chosen to be integer valued. Let us elaborate on this theorem in reference to the separation theorem in convex analysis. Let ρˆ and μ ˆ be the Lov´asz extensions of ρ and μ, respectively. We have ρˆ ≥ μ ˆ on the nonnegative orthant RV+ by the assumption ρ ≥ μ as well as the definition (1.16) of the Lov´asz extension. Define functions g and k by g = ρˆ and k = μ ˆ on RV+ and g = +∞ and k = −∞ elsewhere. Then g is convex and k is concave, by Theorem 1.7, and the separation theorem in convex analysis (Theorem 1.3) applies to the pair of g and k, yielding β ∗ ∈ R and x∗ ∈ RV such that g(p) ≥ β ∗ + p, x∗ ≥ k(p)

(∀ p ∈ RV ).

This inequality for p = χX yields the inequality (1.17) above, where β ∗ = 0 follows from g(0) = ρ(∅) = 0 and k(0) = μ(∅) = 0. Thus, the first half of the discrete separation theorem, the existence of a real vector x∗ , can be proved on the basis of the separation theorem in convex analysis and the relationship between submodularity and convexity. The combinatorial essence of the above theorem, therefore, consists of the second half, claiming the existence of an integer vector for integer-valued functions. Hence, we have the accepted understanding Duality for submodular functions = Convexity + Discreteness, mentioned in section 1.1.1. We denote by S = S[Z] the class of integer-valued submodular set functions and by 0 L = 0 L[Z → Z] that of discrete functions obtained as the restriction to ZV of the Lov´asz extensions of some member of S. That is, 0 L consists of functions g : ZV → Z ∪ {+∞} such that g(p) = ρˆ(p) (∀ p ∈ ZV ) for some ρ ∈ S. In view of the above theorems, 0 L is a promising class of discrete convex functions. This is indeed true, as we will see in section 1.4.1.

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Chapter 1. Introduction to the Central Concepts

1.3.2

Base Polyhedra

A submodular function ρ : 2V → R ∪ {+∞} is associated with a polyhedron B(ρ), called the base polyhedron, defined by B(ρ) = {x ∈ RV | x(X) ≤ ρ(X) (∀ X ⊂ V ), x(V ) = ρ(V )}.

(1.18)

We are particularly interested in the case of integer-valued ρ, for which the base polyhedron is integral in the sense of B(ρ) = B(ρ) ∩ ZV , where the overline designates the convex hull11 in RV . This integrality means, in particular, that all the vertices of the polyhedron B(ρ) are integer points. In this integral case, we refer to B(ρ) as the integral base polyhedron associated with ρ. Assuming the integrality of ρ, we consider a discrete set B = B(ρ) ∩ ZV , the set of integer points contained in integral base polyhedron B(ρ). If integervalued submodular functions can be viewed as well-behaved discrete convex functions, there is a fair chance of such discrete sets B being well-behaved discrete convex sets. This is indeed the case in many senses, as we will see in Chapter 4. Here we focus on an axiomatic characterization of such a B that makes no explicit reference to the defining submodular function ρ. Denoting the positive support and the negative support of a vector x = (x(v) | v ∈ V ) ∈ ZV by supp+ (x) = {v ∈ V | x(v) > 0},

supp− (x) = {v ∈ V | x(v) < 0},

(1.19)

we consider a simultaneous exchange property for a nonempty set B ⊆ ZV : (B-EXC[Z]) For x, y ∈ B and u ∈ supp+ (x − y), there exists v ∈ supp− (x − y) such that x − χu + χv ∈ B and y + χu − χv ∈ B, where χu is the characteristic vector of u ∈ V ; i.e., χu = χ{u} in the notation of (1.14). See Fig. 1.7 for an illustration of this exchange property. The following is a fundamental theorem connecting submodularity and exchangeability.12 Theorem 1.9. The class of integer-valued submodular functions ρ : 2V → Z ∪ {+∞} with ρ(∅) = 0 and ρ(V ) < +∞ and the class of nonempty subsets B ⊆ ZV satisfying (B-EXC[Z]) are in one-to-one correspondence through mutually inverse mappings: ρ → B : B = B(ρ) ∩ ZV , B → ρ : ρ(X) = sup{x(X) | x ∈ B} 11 The

(X ⊆ V ).

convex hull of a set means the smallest convex set containing the set. proofs of Theorems 1.9, 1.10, 1.11, and 1.12 are given later when we come to their rigorous or more general treatments in Theorems 4.15, 8.12, 6.26, and 4.18. 12 The

sidca00si 2013/2/12 page 19

1.3. Submodular Functions and Base Polyhedra

y

-

y + χu − χv = y v 6

19



? x = x − χu + χv 6 

x

- u

Figure 1.7. Exchange property (B-EXC[Z]).

The relationship between submodularity and exchangeability, stated in Theorem 1.9 above, can be reformulated as a conjugacy with respect to the discrete Legendre–Fenchel transformation (1.9). This reformulation establishes a connection to convex analysis. Let M0 [Z] denote the class of nonempty sets B satisfying the exchange axiom ˜ 0 [Z] be the class of the indicator functions δB of B ∈ M0 [Z]; i.e., (B-EXC[Z]) and M M0 [Z] = {B | ∅ = B ⊆ ZV , B satisfies (B-EXC[Z])}, ˜ 0 [Z] = {δB | ∅ = B ⊆ ZV , B satisfies (B-EXC[Z])}, M

(1.20) (1.21)

where δB : ZV → {0, +∞} is defined by  δB (x) =

0 (x ∈ B), +∞ (x ∈ / B).

(1.22)

Recall also the notation 0 L[Z → Z] for the class of the restrictions to ZV of the Lov´ asz extensions of integer-valued submodular set functions. Then Theorem 1.9 can be rewritten as follows. Theorem 1.10. Two classes of discrete functions, 0 L = 0 L[Z → Z] and M0 = ˜ 0 [Z], are in one-to-one correspondence under the discrete Legendre–Fenchel transM formation (1.9). That is, for g ∈ 0 L and f ∈ M0 , we have g • ∈ M0 , f • ∈ 0 L, g •• = g, and f •• = f . The conjugacy relationship between submodularity and exchangeability set forth in the above theorem will be fully generalized to the conjugacy between Lconvexity and M-convexity in the present theory, as will be described soon in section 1.4.3. Fundamental optimization problems on base polyhedra are tractable even under integrality constraints. We consider two representative problems here: 1. the optimal base problem to discuss the issue of local vs. global optimality and

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20

Chapter 1. Introduction to the Central Concepts 2. the (unweighted) intersection problem to show a min-max duality theorem with discreteness assertion.

The two optimization problems on matroids mentioned in section 1.1.1 are special cases of the above problems. This is because the base family of a matroid can be identified, through characteristic vectors of bases, with a nonempty set B of {0, 1}-vectors having the exchange property (B-EXC[Z]). Let B ⊆ ZV be a nonempty set satisfying the exchange axiom (B-EXC[Z]) and c ∈ RV be a given cost (weight) vector. The optimal base problem is to find x ∈ B that minimizes the cost f (x) = c, x = v∈V c(v)x(v). This problem admits the following local optimality criterion for global optimality.13 Theorem 1.11. Assume B ⊆ ZV satisfies (B-EXC[Z]). A point x ∈ B minimizes f (x) = c, x over B if and only if f (x) ≤ f (x − χu + χv ) for all u, v ∈ V such that x − χu + χv ∈ B. To describe the intersection problem we need to introduce another polyhedron P(ρ) = {x ∈ RV | x(X) ≤ ρ(X) (∀ X ⊆ V )},

(1.23)

called the submodular polyhedron, associated with a submodular function ρ : 2V → R ∪ {+∞}. Given a pair of submodular functions ρ1 and ρ2 defined on a common ground set V , the intersection problem is to find a vector x in P(ρ1 ) ∩ P(ρ2 ) that maximizes the sum of the components x(V ). Edmonds’s intersection theorem below shows a min-max duality relation in this problem. Theorem 1.12 (Edmonds’s intersection theorem). Let ρ1 , ρ2 : 2V → R ∪ {+∞} be submodular functions with ρ1 (∅) = ρ2 (∅) = 0, ρ1 (V ) < +∞, and ρ2 (V ) < +∞. Then max{x(V ) | x ∈ P(ρ1 ) ∩ P(ρ2 )} = min{ρ1 (X) + ρ2 (V \ X) | X ⊆ V }.

(1.24)

Moreover, if ρ1 and ρ2 are integer valued, the polyhedron P(ρ1 ) ∩ P(ρ2 ) is integral in the sense of P(ρ1 ) ∩ P(ρ2 ) = P(ρ1 ) ∩ P(ρ2 ) ∩ ZV and there exists an integer-valued vector x∗ that attains the maximum on the lefthand side of (1.24). Discreteness is twofold in Edmonds’s intersection theorem. First, the minimum on the right-hand side of (1.24) is taken over combinatorial objects, i.e., subsets of V , independently of whether the submodular functions are integer valued or not. Second, the maximum can be taken over discrete (integer) points in the case of integer-valued submodular functions. The former is sometimes referred to as the dual integrality and the latter as the primal integrality. 13 This is a generalization of a well-known optimality criterion for the minimum spanning tree problem that a spanning tree is optimal if and only if no improvement is possible by exchanging arcs in and out of the tree. Details are given in Example 6.27.

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1.4. Discrete Convex Functions

21

In sections 1.4.2 and 1.4.4, exchange property (B-EXC[Z]) is generalized to define the concept of M-convex functions and, accordingly, Edmonds’s intersection theorem is generalized to the Fenchel-type duality theorem for M-convex functions.

1.4

Discrete Convex Functions

The backbone of the theory of discrete convex analysis is outlined in this section as a quick preview of the main structural results to be presented in subsequent chapters. The definitions of L-convex and M-convex functions are given, together with concise descriptions of their major properties, including local optimality criteria for global optimality, conjugacy between L-convexity and M-convexity, and various forms of duality theorems. We use the notation dom f = domZ f = {x ∈ ZV | −∞ < f (x) < +∞},

(1.25)

dom g = domR g = {x ∈ RV | −∞ < g(x) < +∞}

(1.26)

for the effective domains of f : ZV → R ∪ {±∞} and g : RV → R ∪ {±∞}.

1.4.1

L-Convex Functions

The first kind of discrete convex functions, L-convex functions, is obtained from a generalization of the Lov´ asz extension of submodular set functions. Let ρ : 2V → R ∪ {+∞} be a submodular set function and ρˆ be its Lov´asz extension, which is indeed an extension of ρ in the sense that ρˆ(χX ) = ρ(X) for X ⊆ V . The submodularity of ρ on 2V , or that of ρˆ on {0, 1}V , extends to the entire space. In fact, it can be shown14 that g = ρˆ satisfies g(p) + g(q) ≥ g(p ∨ q) + g(p ∧ q)

(∀ p, q ∈ RV ),

(1.27)

where p ∨ q and p ∧ q are, respectively, the vectors of componentwise maxima and minima of p and q; i.e., (p ∨ q)(v) = max(p(v), q(v)),

(p ∧ q)(v) = min(p(v), q(v))

(v ∈ V ).

(1.28)

Note that the submodularity inequality (1.13) for ρ is a special case of (1.27) with p = χX and q = χY because of the identities χX∪Y = χX ∨ χY ,

χX∩Y = χX ∧ χY .

(1.29)

It also follows immediately from the definition (1.16) that g(p + α1) = g(p) + αr

(∀ p ∈ RV , ∀ α ∈ R)

(1.30)

for r = ρ(V ), where 1 = (1, 1, . . . , 1) ∈ RV . This shows the linearity of g with respect to the translation of p in the direction of 1. The properties (1.27) and 14 Proofs

of the claims in this subsection are given in Chapter 7.

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22

Chapter 1. Introduction to the Central Concepts p∨q

q qˆ  pˆ p∧q

p

Figure 1.8. Definition of L-convexity.

(1.30) of the Lov´asz extension of a submodular set function are discretized to the following definition of L-convex functions. We say that a function g : ZV → R ∪ {+∞} with domZ g = ∅ is L-convex if it satisfies15,16 (SBF[Z]) (TRF[Z])

g(p) + g(q) ≥ g(p ∨ q) + g(p ∧ q) (∀ p, q ∈ ZV ), ∃ r ∈ R such that g(p + 1) = g(p) + r (∀ p ∈ ZV ).

Naturally, a function k is said to be L-concave if −k is L-convex. Figure 1.8 illustrates, in the case of n = 2, how properties (SBF[Z]) and (TRF[Z]) together can serve as a discrete analogue of convexity. By (SBF[Z]) and (TRF[Z]) we obtain g(p) + g(q) ≥ g(p ∨ q) + g(p ∧ q) = g(ˆ p) + g(ˆ q) for the points pˆ and qˆ, which are discrete approximations to the midpoint (p + q)/2. This inequality may be thought of as a discrete approximation to the midpoint convexity (1.3). We return to midpoint convexity in (1.33) below. It follows from (SBF[Z]) and (TRF[Z]) that the effective domain, say, D, of an L-convex function satisfies17 (SBS[Z]) (TRS[Z])

p, q ∈ D =⇒ p ∨ q, p ∧ q ∈ D, p ∈ D =⇒ p ± 1 ∈ D.

A nonempty set D ⊆ ZV is called L-convex if it satisfies (SBS[Z]) and (TRS[Z]) above. Obviously, a set D is L-convex if and only if its indicator function δD is an L-convex function. Since an L-convex function g is linear in the direction of 1, we may dispense with this direction as far as we are interested in its nonlinear behavior. Namely, instead of the function g in n = |V | variables, we may consider a function g  in n − 1 variables defined by (1.31) g  (p ) = g(0, p ), 15 SBF 16 The

stands for submodularity for functions and TRF for translation for functions. notation ∃ means “there exists” or “for some” in contrast to ∀ meaning “for all” or “for

any.” 17 SBS stands for submodularity for sets and TRS for translation for sets.

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1.4. Discrete Convex Functions

p+q

23

q q

p+q

2

p+q p

2

p+q

2

2

p+q 2

q

p+q p

p

2

Figure 1.9. Discrete midpoint convexity.

where, for an arbitrarily fixed element v0 ∈ V , a vector p ∈ ZV is represented as  p = (p0 , p ), with p0 = p(v0 ) ∈ Z and p ∈ ZV for V  = V \ {v0 }. Note that the effective domain domZ g  of g  is the restriction of domZ g to the coordinate plane defined by p0 = 0. A function g  derived from an L-convex function by such a restriction is called an L -convex18 function. More formally, an L -convex function is defined as follows. Let 0 denote a new element not in V and put V˜ = {0} ∪ V . A function g : ZV → R ∪ {+∞} is called ˜ L -convex if the function g˜ : ZV → R ∪ {+∞} defined by g˜(p0 , p) = g(p − p0 1)

(p0 ∈ Z, p ∈ ZV )

(1.32)

is L-convex. It turns out that L -convexity can be characterized by a kind of generalized submodularity: (SBF [Z]) g(p) + g(q) ≥ g((p − α1) ∨ q) + g(p ∧ (q + α1)) (∀ p, q ∈ ZV , ∀ α ∈ Z+ ), which we name translation submodularity. Note that this inequality for α = 0 coincides with the original submodularity (SBF[Z]). An alternative characterization of L -convexity is by discrete midpoint convexity (see Fig. 1.9):     p+q p+q (1.33) g(p) + g(q) ≥ g +g (p, q ∈ ZV ), 2 2

and p+q denote, respectively, the integer vectors obtained from p+q where p+q 2 2 2 by componentwise round-up and round-down to the nearest integers. The discrete midpoint convexity is a natural approximation to the midpoint convexity (1.3) of ordinary convex functions. Whereas L -convex functions are conceptually equivalent to L-convex functions, the class of L -convex functions is strictly larger than that of L-convex functions. In fact, it is easy to derive the translation submodularity (SBF [Z]) from (SBF[Z]) and (TRF[Z]) or, more intuitively, a comparison of Figs. 1.9 and 1.8 indicates this. The simplest example of an L -convex function that is not L-convex is the one-dimensional discrete convex function depicted in Fig. 1.6 (left). 18 “L -convex”

should be read “L-natural-convex.”

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Chapter 1. Introduction to the Central Concepts

L-convex functions enjoy the following nice properties that are expected of discrete convex functions. • An L-convex function can be extended to a convex function. • Local optimality (or minimality) guarantees global optimality. Specifically, we have the following: – For an L-convex function g and a point p ∈ domZ g,  g(p) ≤ g(p + χX ) (∀ X ⊆ V ), V g(p) ≤ g(q) (∀ q ∈ Z ) ⇐⇒ g(p) = g(p + 1). – For an L -convex function g and a point p ∈ domZ g, g(p) ≤ g(q) (∀ q ∈ ZV ) ⇐⇒ g(p) ≤ g(p ± χX )

(∀ X ⊆ V ).

Thus L-convex functions are endowed with the property of discreteness in direction. • Discrete duality, e.g., the Fenchel-type min-max duality or discrete separation, holds good. Thus, L-convex functions are endowed with the property of discreteness in value. (This will be explained in section 1.4.4.) • Efficient algorithms can be designed for the minimization of an L-convex function and for the Fenchel-type min-max duality. L-convexity is closely related to network flow problems such as the minimum cost flow problem and the shortest path problem. As an indication of this connection we mention that, given an integer-valued distance function19 γ on V , the set of admissible integer-valued potentials D = {p ∈ ZV | p(v) − p(u) ≤ γ(u, v) (∀ u, v ∈ V, u = v)}

(1.34)

is an L-convex set. The converse is also true; i.e., any L-convex set has such a polyhedral description for some γ satisfying the triangle inequality γ(u, v) + γ(v, w) ≥ γ(u, w)

(u, v, w ∈ V ).

(1.35)

The concepts of L-/L -convexity can also be defined for functions in real variables through an appropriate adaptation of the conditions (SBF[Z]) and (TRF[Z]). Namely, we can define a function g : RV → R ∪ {+∞} with domR g = ∅ to be L-convex if (SBF[R]) g(p) + g(q) ≥ g(p ∨ q) + g(p ∧ q) (∀ p, q ∈ RV ), (TRF[R]) ∃ r ∈ R such that g(p + α1) = g(p) + αr (∀ p ∈ RV , ∀ α ∈ R). 19 An integer-valued distance function on V means a function γ : V × V → Z ∪ {+∞} such that γ(v, v) = 0 for all v ∈ V .

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1.4. Discrete Convex Functions

25

L -convex functions are defined as the restriction of L-convex functions, as in (1.31), and are characterized by (SBF [R]) g(p) + g(q) ≥ g((p − α1) ∨ q) + g(p ∧ (q + α1)) (∀ p, q ∈ RV , ∀ α ∈ R+ ). More precisely, L-convexity can be defined for closed proper convex functions.20 Instead of dealing with this most general class of functions, this monograph focuses asz extension on polyhedral convex functions21 and quadratic functions. The Lov´ of a submodular set function is a polyhedral L-convex function that has the additional property of being positively homogeneous. Quadratic L -convex functions are characterized in section 2.1.2 as quadratic forms defined by diagonally dominant symmetric M-matrices22 and hence they are equivalent to the (finite-dimensional) Dirichlet forms known in probability theory. We conclude this section by identifying the four types of L-convex functions that we are concerned with: real-valued L-convex functions on integers, integervalued L-convex functions on integers, real-valued polyhedral L-convex functions on reals, and quadratic L-convex functions on reals. For the first three classes we introduce the following notation: L[Z → R] = {g : ZV → R ∪ {+∞} | g is L-convex},

(1.36)

L[Z → Z] = {g : Z → Z ∪ {+∞} | g is L-convex}, L[R → R] = {g : RV → R ∪ {+∞} | g is polyhedral L-convex}.

(1.37) (1.38)

V

Note the inclusion 0 L[Z

→ Z] ⊆ L[Z → Z] ⊆ L[Z → R],

where 0 L[Z → Z] is the notation from section 1.3.1 for the class of the restrictions to ZV of the Lov´asz extensions of integer-valued submodular set functions.

1.4.2

M-Convex Functions

The second kind of discrete convex functions, M-convex functions, is obtained from a generalization of the simultaneous exchange property (B-EXC[Z]) of base polyhedra. As a motivation for the axiom of M-convex functions, let us first observe that a convex function f : Rn → R ∪ {+∞} satisfies the inequality f (x) + f (y) ≥ f (x − α(x − y)) + f (y + α(x − y))

(1.39)

for every α with 0 ≤ α ≤ 1. The validity of this inequality can be verified easily from the definition of a convex function by adding the inequality (1.2) for λ = α and (1.2) for λ = 1 − α. 20 The

definition of closed proper convex function can be found in section 3.1. polyhedral convex function is a function that can be represented as the maximum of a finite number of affine functions on a polyhedral effective domain. 22 Here is an unfortunate conflict of our notation with the standard terminology in matrix theory. M-matrices do not correspond to M-convex functions but to L-convex functions. 21 A

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Chapter 1. Introduction to the Central Concepts

6

y

~ y

f (x)

x − α(x − y)



y + α(x − y)

x }

x y

y

y + α(x − y)

 x

x

x − α(x − y)

Figure 1.10. Property of a convex function.

The inequality (1.39) above shows that the sum of the function values evaluated at two points, x and y, does not increase if the two points approach each other by the same distance on the line segment connecting them (see Fig. 1.10). For a function defined on discrete points Zn , we simulate this property by moving two points along the coordinate axes rather than on the connecting line segment. We say that a function f : ZV → R ∪ {+∞} with domZ f = ∅ is M-convex if it satisfies the following exchange axiom: (M-EXC[Z]) For x, y ∈ domZ f and u ∈ supp+ (x − y), there exists v ∈ supp− (x − y) such that f (x) + f (y) ≥ f (x − χu + χv ) + f (y + χu − χv ).

(1.40)

See Fig. 1.11 for an illustration of this exchange property. The inequality (1.40) implicitly imposes the condition that x−χu +χv ∈ domZ f and y +χu −χv ∈ domZ f for the finiteness of the right-hand side. With the use of the notation Δf (z; v, u) = f (z + χv − χu ) − f (z)

(1.41)

for z ∈ domZ f and u, v ∈ V , the exchange axiom (M-EXC[Z]) can be expressed alternatively as follows: (M-EXC [Z]) For x, y ∈ domZ f , max

min

[Δf (x; v, u) + Δf (y; u, v)] ≤ 0,

u∈supp+ (x−y) v∈supp− (x−y)

(1.42)

where the maximum and the minimum over an empty set are −∞ and +∞, respectively. Naturally, a function h is said to be M-concave if −h is M-convex. It follows from (M-EXC[Z]) that the effective domain of an M-convex function satisfies the exchange axiom (B-EXC[Z]) that characterizes the set of integer points in an integral base polyhedron, since x−χu +χv ∈ domZ f and y +χu −χv ∈ domZ f

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1.4. Discrete Convex Functions

27 -

y

y + χu − χv = y v 6

?



x = x − χu + χv 6 

x

- u

Figure 1.11. Exchange property in the definition of M-convexity.

for x, y ∈ domZ f in (1.40). In particular, the indicator function δB : ZV → {0, +∞} of a set B ⊆ ZV is M-convex if and only if B is the set of integer points in an integral base polyhedron. Accordingly, we refer to a set of integer points satisfying (B-EXC[Z]) as an M-convex set . The effective domain of an M-convex function f , being an M-convex set, lies on a hyperplane {x ∈ RV | x(V ) = r} for some integer r and, accordingly, we may consider the projection of f along a coordinate axis. This means that, instead of the function f in n = |V | variables, we consider a function f  in n − 1 variables defined by (1.43) f  (x ) = f (x0 , x ) with x0 = r − x (V  ), where V  = V \ {v0 } for an arbitrarily fixed element v0 ∈ V and a vector x ∈ ZV  is represented as x = (x0 , x ) with x0 = x(v0 ) ∈ Z and x ∈ ZV . Note that the effective domain domZ f  of f  is the projection of domZ f along the chosen coordinate axis v0 . A function f  derived from an M-convex function by such a projection is called an M -convex23 function. More formally, an M -convex function is defined as follows. Let 0 denote a new element not in V and put V˜ = {0} ∪ V . A function f : ZV → R ∪ {+∞} is ˜ called M -convex if the function f˜ : ZV → R ∪ {+∞} defined by  f (x) if x0 = −x(V ), (1.44) (x0 ∈ Z, x ∈ ZV ) f˜(x0 , x) = +∞ otherwise is an M-convex function. It turns out24 that an M -convex function f can be characterized by a similar exchange property: (M -EXC[Z]) For x, y ∈ domZ f and u ∈ supp+ (x − y),  f (x) + f (y) ≥ min f (x − χu ) + f (y + χu ), min

 {f (x − χu + χv ) + f (y + χu − χv )} .

v∈supp− (x−y) 23 “M -convex” 24 Proofs

should be read “M-natural-convex.” of the claims in this subsection are given in Chapter 6.

(1.45)

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Chapter 1. Introduction to the Central Concepts

Whereas M -convex functions are conceptually equivalent to M-convex functions, the class of M -convex functions is strictly larger than that of M-convex functions. This follows from the implication (M-EXC[Z]) ⇒ (M -EXC[Z]). The simplest example of an M -convex function that is not M-convex is the one-dimensional discrete convex function depicted in Fig. 1.6 (left). M-convex functions enjoy the following nice properties that are expected of discrete convex functions. • An M-convex function can be extended to a convex function. • Local optimality (or minimality) guarantees global optimality. Specifically, we have the following: – For an M-convex function f and a point x ∈ domZ f , f (x) ≤ f (y) (∀ y ∈ ZV ) ⇐⇒ f (x) ≤ f (x − χu + χv ) (∀ u, v ∈ V ). (This is a generalization of Theorem 1.11.) – For an M -convex function f and a point x ∈ domZ f ,  f (x) ≤ f (y) (∀ y ∈ ZV ) ⇐⇒

f (x) ≤ f (x − χu + χv ) (∀ u, v ∈ V ), f (x) ≤ f (x ± χv ) (∀ v ∈ V ).

Thus, M-convex functions are endowed with the property of discreteness in direction. • Discrete duality, e.g., the Fenchel-type min-max duality or discrete separation, holds good. Thus, M-convex functions are endowed with the property of discreteness in value. (This will be explained in section 1.4.4.) • Efficient algorithms can be designed for the minimization of an M-convex function and for the Fenchel-type min-max duality. M-convex functions are closely related to network flow problems, such as the minimum cost flow problem and the shortest path problem. As an indication of this connection we mention that the distance γ on V defined by γ(u, v) = Δf (x; v, u) for u, v ∈ V with a fixed x ∈ domZ f satisfies the triangle inequality (1.35). This is because the exchange property (M-EXC[Z]) applied to x ˜ = x − χv + χw and y˜ = x − χu + χv , for which supp+ (˜ x − y˜) = {u, w} and supp− (˜ x − y˜) = {v}, y + χu − χv ), which is equivalent to yields f (˜ x) + f (˜ y ) ≥ f (˜ x − χu + χv ) + f (˜ Δf (x; w, v) + Δf (x; v, u) ≥ Δf (x; w, u). The concepts of M-/M -convexity can also be defined for functions in real variables through an appropriate adaptation of the exchange axiom. Namely, we can define a function f : RV → R ∪ {+∞} with domR f = ∅ to be M-convex if it satisfies the following exchange property:

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29

(M-EXC[R]) For x, y ∈ domR f and u ∈ supp+ (x − y), there exist v ∈ supp− (x − y) and a positive number α0 ∈ R++ such that f (x) + f (y) ≥ f (x − α(χu − χv )) + f (y + α(χu − χv )) for all α ∈ R with 0 ≤ α ≤ α0 . M -convex functions are defined as the projection of M-convex functions, as in (1.43), and are characterized by the following: (M -EXC[R]) For x, y ∈ domR f and u ∈ supp+ (x − y), there exist v ∈ supp− (x − y) ∪ {0} and a positive number α0 ∈ R++ such that f (x) + f (y) ≥ f (x − α(χu − χv )) + f (y + α(χu − χv )) for all α ∈ R with 0 ≤ α ≤ α0 , where χ0 = 0 by convention. More precisely, M-convexity can be defined for closed proper convex functions. Instead of dealing with this most general class of functions, this monograph focuses on polyhedral convex functions and quadratic functions. Polyhedral M-convex functions are a quantitative generalization of the base polyhedra explained in section 1.3.2, whereas quadratic M-convex functions are characterized in section 2.1.3 as quadratic forms defined by the inverse of diagonally dominant symmetric M-matrices. We conclude this section by identifying the four types of M-convex functions that we are concerned with: real-valued M-convex functions on integers, integervalued M-convex functions on integers, real-valued polyhedral M-convex functions on reals, and quadratic M-convex functions on reals. For the first three classes we introduce the following notation: M[Z → R] = {f : ZV → R ∪ {+∞} | f is M-convex},

(1.46)

M[Z → Z] = {f : Z → Z ∪ {+∞} | f is M-convex}, M[R → R] = {f : RV → R ∪ {+∞} | f is polyhedral M-convex}.

(1.47) (1.48)

V

Note the inclusion ˜ 0 [Z] ⊆ M[Z → Z] ⊆ M[Z → R], M ˜ 0 [Z] is the notation from section 1.3.2 for the class of indicator functions where M of sets of integer points contained in integral base polyhedra.

1.4.3

Conjugacy

The conjugacy relationship between L-convexity and M-convexity is a distinguishing feature of the present theory. Whereas conjugacy in ordinary convex analysis gives a symmetric one-to-one correspondence within a single class of closed proper convex functions (Theorem 1.2), conjugacy described in this section establishes a one-to-one correspondence between two different classes of discrete functions having different combinatorial properties denoted by “L” and “M.” We describe the

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Chapter 1. Introduction to the Central Concepts

conjugacy for integer-valued L-convex and M-convex functions on integer points, namely, for L = L[Z → Z] and M = M[Z → Z], although a similar conjugacy relationship exists between L -convex and M -convex functions and also between their polyhedral versions. In Theorem 1.10 we saw the conjugacy between 0 L = 0 L[Z → Z] and M0 = ˜ 0 [Z] as a reformulation of the equivalence between submodularity for set functions M and exchangeability for discrete sets stated in Theorem 1.9. Since 0 L and M0 are subclasses of L and M, respectively, we can summarize our present knowledge as L ⊇

0L

←→ M0 ⊆ M,

where ←→ above denotes the conjugacy with respect to the discrete Legendre– Fenchel transformation (1.9). The following theorem25 shows that the conjugacy extends to a relation between L and M. Theorem 1.13 (Discrete conjugacy theorem). The classes of integer-valued Lconvex functions and M-convex functions, L = L[Z → Z] and M = M[Z → Z], are in one-to-one correspondence under the discrete Legendre–Fenchel transformation (1.9). That is, for g ∈ L and f ∈ M, we have g • ∈ M, f • ∈ L, g •• = g, and f •• = f . The essence of the relationship between M0 and 0 L is the conjugacy between M-convex sets and their support functions, the latter being positively homogeneous L-convex functions. Symmetrically, we can formulate the conjugacy between Lconvex sets and their support functions in the following theorem, where we denote by L˜0 [Z] the class of the indicator functions of L-convex sets and by 0 M[Z → Z] that of positively homogeneous M-convex functions. Theorem 1.14. Two classes of discrete functions, L0 = L˜0 [Z] and 0 M = 0 M[Z → Z], are in one-to-one correspondence under the discrete Legendre–Fenchel transformation (1.9). That is, for g ∈ L0 and f ∈ 0 M, we have g • ∈ 0 M, f • ∈ L0 , g •• = g, and f •• = f . Just as a positively homogeneous L-convex function can be identified with a submodular set function, so can a positively homogeneous M-convex function f be identified with a distance function γ on V satisfying the triangle inequality (1.35). The correspondence is given by γ(u, v) = f (χv − χu )

(u, v ∈ V ),

which establishes a one-to-one mapping between 0 M and T , where T = T [Z] denotes the class of distance functions on V satisfying the triangle inequality (1.35). Figure 1.12 demonstrates the conjugacy relations as well as the one-to-one correspondences explained in the above. This diagram clarifies the relationship among various classes of combinatorial objects, including submodular functions 25 The

proofs of Theorems 1.13 and 1.14 are given in Theorem 8.12 and (8.17), respectively.

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31

M-convex functions positively homogeneous M-convex functions (Theorem 6.59) distance functions (Theorem 5.5) L-convex sets

0M



T L0



M  ⏐   ⏐  L

⊃ M0 S ⊃

0L

L-convex functions

(base polyhedra) M-convex sets (Theorem 4.15) submodular set functions (Theorem 7.40) positively homogeneous L-convex functions

Figure 1.12. Conjugacy in discrete convexity. (S), distance functions (T ), and base polyhedra (M0 ). It is recalled again that the conjugacy is defined by the (discrete) Legendre–Fenchel transformation (1.9). The pair of L- and M-convexity prevails in discrete systems. • In network flow problems, flow and tension are dual objects. Roughly speaking, flow corresponds to M-convexity and tension to L-convexity. Namely, tension : L ←→ M : flow. In multiterminal electrical networks consisting of nonlinear resistors, the equilibrium state can be characterized as a stationary point of a convex function representing the energy (or power). The function is M-convex when expressed in terms of the terminal current supplied by current sources. It is L-convex when expressed in terms of the terminal voltage (or potential) specified by voltage sources. Network flow problems are discussed in section 2.2 and Chapter 9. • In a matroid, the rank function corresponds to L-convexity and the base family to M-convexity: rank function : L ←→ M : base family. In a valuated matroid, the valuation of bases is an M-concave function defined on the unit cube {0, 1}V . Matroids and valuated matroids are explained in section 2.4. • The concept of M-matrices corresponds to L-convexity. Specifically, a quadratic function is L -convex if and only if it is defined by a diagonally dominant symmetric M-matrix. The inverse of such a matrix corresponds to M-convexity. A diagonally dominant symmetric M-matrix arises, for instance, from a discretization of the Poisson problem of partial differential equations, where the matrix is an approximation to the differential operator (Laplacian) and its inverse corresponds to the Green function. Hence differential operator : L ←→ M : Green function.

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32

Chapter 1. Introduction to the Central Concepts Dirichlet forms in probability theory are exactly the same as L -convex quadratic functions. These quadratic forms are discussed in section 2.1.

1.4.4

Duality

Duality theorems for L- and M-convex functions are stated here in the case of integer-valued functions defined on integer points.26 We explain their significance in relation to previous results, such as Frank’s discrete separation theorem for submodular/supermodular functions, Edmonds’s intersection theorem, and Frank’s weightsplitting theorem for the weighted matroid intersection problem. Recall from section 1.2 the generic form of a Fenchel-type min-max duality theorem: [Discrete Fenchel-type duality theorem] Let f : ZV → Z ∪ {+∞} and h : ZV → Z ∪ {−∞} be convex and concave functions, respectively (in an appropriate sense). Then min{f (x) − h(x) | x ∈ ZV } = max{h◦ (p) − f • (p) | p ∈ ZV }. We can now specify the meaning of convexity left open in this generic statement by L-convexity or M-convexity. Then the following theorems result. Theorem 1.15 (Fenchel-type duality for L-convex functions). Let g : ZV → Z ∪ {+∞} be an L -convex function and k : ZV → Z ∪ {−∞} be an L -concave function such that domZ g ∩ domZ k = ∅ or domZ g • ∩ domZ k ◦ = ∅. Then we have inf{g(p) − k(p) | p ∈ ZV } = sup{k ◦ (x) − g • (x) | x ∈ ZV }.

(1.49)

If this common value is finite, the infimum is attained by some p ∈ domZ g ∩ domZ k and the supremum is attained by some x ∈ domZ g • ∩ domZ k ◦ . Theorem 1.16 (Fenchel-type duality for M-convex functions). Let f : ZV → Z ∪ {+∞} be an M -convex function and h : ZV → Z ∪ {−∞} be an M -concave function such that domZ f ∩ domZ h = ∅ or domZ f • ∩ domZ h◦ = ∅. Then we have inf{f (x) − h(x) | x ∈ ZV } = sup{h◦ (p) − f • (p) | p ∈ ZV }.

(1.50)

If this common value is finite, the infimum is attained by some x ∈ domZ f ∩ domZ h and the supremum is attained by some p ∈ domZ f • ∩ domZ h◦ . Although the above theorems look different, they are actually the same theorem if we assume the conjugacy between L -convex functions and M -convex functions (a variant of Theorem 1.13). In fact, substitution of g = f • and k = h◦ in (1.49) yields inf{f • (p) − h◦ (p) | p ∈ ZV } = sup{(h◦ )◦ (x) − (f • )• (x) | x ∈ ZV }, 26 The proofs of Theorems 1.15, 1.16, 1.17, 1.18, 1.23, and 1.24 are given in Theorems 8.21, 8.21, 8.16, 8.15, 5.9, and 4.21, respectively.

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33

which is equivalent to (1.50) by (f • )• = f and (h◦ )◦ = h. Thus, the Fenchel-type min-max theorem is self-conjugate. Next we turn to discrete separation theorems. Recall, again from section 1.2, the generic form of a discrete separation theorem: [Discrete separation theorem] Let f : ZV → Z∪{+∞} and h : ZV → Z∪ {−∞} be convex and concave functions, respectively (in an appropriate sense). If f (x) ≥ h(x) (∀ x ∈ ZV ), there exist α∗ ∈ Z and p∗ ∈ ZV such that f (x) ≥ α∗ + p∗ , x ≥ h(x)

(∀ x ∈ ZV ).

We can substitute L-convexity or M-convexity for convexity in this generic statement to obtain a conjugate pair of discrete separation theorems. Theorem 1.17 (L-separation theorem). Let g : ZV → Z ∪ {+∞} be an L V convex function and k : Z → Z ∪ {−∞} be an L -concave function such that domZ g ∩ domZ k = ∅ or domZ g • ∩ domZ k ◦ = ∅. If g(p) ≥ k(p) (∀ p ∈ ZV ), there exist β ∗ ∈ Z and x∗ ∈ ZV such that g(p) ≥ β ∗ + p, x∗ ≥ k(p)

(∀ p ∈ ZV ).

(1.51)

Theorem 1.18 (M-separation theorem). Let f : ZV → Z ∪ {+∞} be an M convex function and h : ZV → Z ∪ {−∞} be an M -concave function such that domZ f ∩ domZ h = ∅ or domZ f • ∩ domZ h◦ = ∅. If f (x) ≥ h(x) (∀ x ∈ ZV ), there exist α∗ ∈ Z and p∗ ∈ ZV such that f (x) ≥ α∗ + p∗ , x ≥ h(x)

(∀ x ∈ ZV ).

(1.52)

These duality theorems include a number of previous important results as special cases. We demonstrate this for Frank’s discrete separation theorem for submodular/supermodular functions, Edmonds’s intersection theorem, and Frank’s weight-splitting theorem for the weighted matroid intersection problem. Example 1.19. Frank’s discrete separation theorem (Theorem 1.8) in the integral case can be derived from the L-separation theorem (Theorem 1.17). The submodular and supermodular functions ρ and μ can be identified, respectively, with an L -convex function g : ZV → Z ∪ {+∞} and an L -concave function k : ZV → Z ∪ {−∞} by ρ(X) = g(χX ) and μ(X) = k(χX ) for X ⊆ V , where domZ g ⊆ {0, 1}V and domZ k ⊆ {0, 1}V . The L-separation theorem applies, since the first assumption, domZ g ∩ domZ k = ∅, is met by g(0) = k(0) = 0, which follows from ρ(∅) = μ(∅) = 0. We see β ∗ = 0 from the inequality (1.51) for p = 0, and then the desired inequality (1.17) is obtained from (1.51) with p = χX for X ⊆ V .

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Chapter 1. Introduction to the Central Concepts

Example 1.20. Edmonds’s intersection theorem (Theorem 1.12) in the integral case, max{x(V ) | x ∈ P(ρ1 ) ∩ P(ρ2 ) ∩ ZV } = min{ρ1 (X) + ρ2 (V \ X) | X ⊆ V },

(1.53)

can be derived from the Fenchel-type duality theorem for M-convex functions (Theorem 1.16). Define f (x) = δ1 (x) and h(x) = 1, x − δ2 (x) by using the indicator functions δi (x) of P(ρi ) ∩ ZV (i = 1, 2). Then f is M -convex and h is M -concave with domZ f ∩ domZ h = ∅. An easy calculation yields f • (p) =

sup p, x ,

h◦ (p) = − sup 1 − p, x ,

x∈P(ρ1 )

x∈P(ρ2 )

which implies domZ f • ∩ domZ h◦ ⊆ {0, 1}V and f • (p) = ρ1 (X),

h◦ (p) = −ρ2 (V \ X)

(p = χX , X ⊆ V ).

Substituting these expressions into the Fenchel-type min-max relation (1.50) yields inf{δ1 (x) − 1, x + δ2 (x) | x ∈ ZV } = sup{−ρ2 (V \ X) − ρ1 (X) | X ⊆ V }, which is equivalent to the desired equation (1.53). Example 1.21. Frank’s weight-splitting theorem for the matroid intersection problem with integer weights is a special case of the M-separation theorem (Theorem 1.18). Given two matroids (V, B1 ) and (V, B2 ) on a common ground set V with base families B1 and B2 , as well as an integer-valued weight vector w : V → Z, the optimal common base problem is to find B ∈ B1 ∩ B2 that minimizes the weight w(B) = v∈B w(v). Frank’s weight-splitting theorem says that a common base B ∗ ∈ B1 ∩ B2 is optimal if and only if there exist integer vectors w1∗ and w2∗ such that (i) w = w1∗ + w2∗ , (ii) B ∗ is a minimum-weight base of (V, B1 ) with respect to w1∗ , and (iii) B ∗ is a minimum-weight base of (V, B2 ) with respect to w2∗ . The “if” part is easy and the content of this theorem lies in the assertion about the existence of such a weight splitting. For an optimal common base B ∗ , define   w(B) (x = χB , B ∈ B1 ), w(B ∗ ) (x = χB , B ∈ B2 ), f (x) = h(x) = +∞ (otherwise), −∞ (otherwise), which are M-convex and M-concave, respectively (h is constant on B2 ). Noting that f (x) ≥ h(x) (x ∈ ZV ), as well as domZ f ∩ domZ h = ∅, we apply the M-separation theorem to obtain α∗ ∈ Z and p∗ ∈ ZV for which the inequality (1.52) is true. A weight splitting constructed by w1∗ = w − p∗ ,

w2∗ = p∗

has the desired properties (i) to (iii). In fact, (1.52) with x = χB ∗ reads w(B ∗ ) ≥ α∗ + p∗ (B ∗ ) ≥ w(B ∗ ), which shows α∗ = w(B ∗ ) − p∗ (B ∗ ) = w1∗ (B ∗ ). It follows

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35

M-separation theorem f (x) ≥ α∗ + p∗ , x ≥ h(x) Fenchel-type duality (Fujishige [62]) ! Intersection theorem (Edmonds [44]) ⎧ ! ⎪ ⎪ ⎪ ⇒ Discrete separation for submodular functions Fenchel-type duality ⎪ ⎨ (Frank [55]) inf{f − h} ⎪ ⎪ ⇒ Valuated matroid intersection = sup{h◦ − f • } ⎪ ⎪ ⎩ (Murota [135]) ! ⇓ L-separation theorem Weighted matroid intersection f • (p) ≥ β ∗ + p, x∗ ≥ h◦ (p) (Edmonds [45], Frank [54], Iri–Tomizawa [96]) Figure 1.13. Duality theorems (f : M -convex function, h: M -concave function).

S1 S2 p∗ k

Figure 1.14. Separation for convex sets.

also from (1.52) that w(B) ≥ α∗ + p∗ (B) for every B ∈ B1 (namely, (ii)) and α∗ + p∗ (B) ≥ w(B ∗ ) for every B ∈ B2 (namely, (iii)). Moreover, the valuated matroid intersection theorem, a generalization of the weight-splitting theorem, can be regarded as a special case of the M-separation theorem (to be explained in Example 8.28). The relationship among duality theorems is summarized in Fig. 1.13. A derivation of Fujishige’s Fenchel-type duality theorem from the Fenchel-type duality theorem for L-convex functions will be explained in Example 8.26. We conclude this section with discrete separation theorems for a pair of Lconvex sets and for a pair of M-convex sets. First we recall the separation theorem for a pair of convex sets (see Fig. 1.14). Theorem 1.22 (Separation for convex sets). If S1 and S2 are disjoint convex sets

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Chapter 1. Introduction to the Central Concepts

in RV , there exists a nonzero vector p∗ ∈ RV such that inf{ p∗ , x | x ∈ S1 } − sup{ p∗ , x | x ∈ S2 } ≥ 0. The discrete versions of this theorem read as follows. Theorem 1.23 (Discrete separation for L-convex sets). If D1 and D2 are disjoint L-convex sets, there exists x∗ ∈ {−1, 0, 1}V such that inf{ p, x∗ | p ∈ D1 } − sup{ p, x∗ | p ∈ D2 } ≥ 1.

(1.54)

Theorem 1.24 (Discrete separation for M-convex sets). If B1 and B2 are disjoint M-convex sets, there exists p∗ ∈ {0, 1}V ∪ {0, −1}V such that inf{ p∗ , x | x ∈ B1 } − sup{ p∗ , x | x ∈ B2 } ≥ 1.

(1.55)

Let us dwell on the content of these theorems, referring to the latter. The first implication, explicit in the statement of Theorem 1.24, is that the separating vector p∗ is so special that p∗ or −p∗ is a {0, 1}-vector. The second, less conspicuous and more subtle, is that B1 ∩ B2 = ∅ follows from B1 ∩ B2 = ∅, since (1.55) implies B1 ∩ B2 = ∅. The implication B1 ∩ B2 = ∅ =⇒ B1 ∩ B2 = ∅ for a pair of discrete sets comprises an essential ingredient in a successful theory of discrete convexity, as will be discussed in section 3.3.

1.4.5

Classes of Discrete Convex Functions

Besides L-, M-, L -, and M -convex functions, we will consider in this book some other classes of discrete convex functions, including integrally convex functions, L2 -convex functions, and M2 -convex functions, whose definitions are given later. The inclusion relationships among these classes of discrete convex functions are depicted in Fig. 1.15 for ease of reference. The properties of these discrete convex functions with respect to various fundamental operations are summarized in Table 1.2; counterexamples for the failure of the properties can be found in Murota– Shioura [153].

Bibliographical Notes References for optimization abound in the literature. See, e.g., Nemhauser–Rinnooy Kan–Todd [166] as a general handbook; Bazaraa–Sherali–Shetty [8], Bertsekas [10], Fletcher [52], Mangasarian [126], and Nocedal–Wright [169] for nonlinear optimization; and Cook–Cunningham–Pulleyblank–Schrijver [26], Du–Pardalos [43], Korte– Vygen [115], Lawler [119], and Nemhauser–Wolsey [167] for combinatorial optimization. References for convex analysis are included in the bibliographical notes at the

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37

f : ZV → R ∪ {+∞} Miller’s discrete convex

integrally convex M2 -convex M -convex separable convex L -convex L2 -convex convex-extensible

=

Figure 1.15. Classes of discrete convex functions (M -convex ∩ L -convex ∩ L2 -convex = separable convex).

M2 -convex

end of Chapter 3, and those for network flow theory and matroid theory are in Chapter 2. Fujishige [65] is a standard reference for submodular functions, and Narayanan [165] and Topkis [203] cover some other topics related to electrical networks and economics, respectively. Theorem 1.7, connecting submodularity and convexity, is due to Lov´ asz [123], and the name “Lov´ asz extension” was coined by Fujishige [63], [65]. The discrete separation for submodular functions, Theorem 1.8, is due to Frank [55]. Theorem 1.9, the equivalence between submodularity and exchangeability, is folklore from the 1980s. Seeing that no explicit and rigorous proof can be found in the literature, we will provide a proof in Theorem 4.15 in this book. The recasting into Theorem 1.10 is by Murota [140]. The local optimality criterion for the linearly weighted base problem in a matroid (Theorem 1.11) is a standard result (see, e.g., Corollary 8.7 of [26]). The intersection theorem, Theorem 1.12, is due to Edmonds [44]. The weight-splitting theorem described in Example 1.21 is due to Frank [54]. M-convex functions are introduced in Murota [137], followed by L-convex functions in Murota [140]. Their fundamental properties are established in Murota [137], [140], [141], [142]; the discrete conjugacy theorem (Theorem 1.13) and the L-separation theorem (Theorem 1.17) in [140]; the M-separation theorem (Theorem 1.18) in [137], [140], [142]; and the Fenchel-type duality theorem for M-convex functions (Theorem 1.16) in [137], [140]. The separation theorems for L-convex sets and M-convex sets (Theorems 1.23 and 1.24) are due to [140]. M-convexity

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Chapter 1. Introduction to the Central Concepts

Table 1.2. Operations for discrete convex sets and functions (f : function, S: set; #: Yes [cf. Theorem, Prop.], ×: No).

f1 + f2 S1 ∩ S2 f + sep-conv S ∩ [a, b] f + affine f1 2Z f2 S1 + S2 f• dom f arg min f

Miller’s discrete convex × × × # × × × × # # M2 -convex

convex extensible # # # # # × × # # #

integrally convex × × # [3.24] # # [3.25] × × × # [3.28] # [3.28]

separable convex # # # # # # # # # #

L2 -convex

M -convex

L -convex

f1 + f2 S1 ∩ S2 f + sep-conv S ∩ [a, b] f + affine

× × # # #

× × × × #

× × # # #

(M2 -conv) (M2 -conv)

[6.15]

# # # # #

f1 2Z f2 S1 + S2

× ×

× ×

# #

[6.15] [4.23]

× (L2 -conv) × (L2 -conv)

× (L2 -conv) # [8.29] # [8.30]

× (M2 -conv) # [8.39] # [8.40]

× (L -conv) # [6.7] # [6.29]

f• dom f arg min f

[6.15]

[7.11] [5.7] [7.11] [7.11]

× (M -conv) # [7.8] # [7.16]

sep-conv: separable convex function, affine: affine function, dom : effective domain (1.25), arg min: set of minimizers (3.16), 2Z : integer infimal convolution (6.43), f • : conjugate (1.9) of integer-valued function f and L-convexity are investigated also for functions in real variables for polyhedral, quadratic, and closed convex functions in Murota–Shioura [152], [155], [156], [157]. M -convex functions are introduced by Murota–Shioura [151] and L -convex functions by Fujishige–Murota [68]. The concept of submodular integrally convex functions, together with a characterization by discrete midpoint convexity, is due to Favati–Tardella [49]. The equivalence of this concept to L -convexity is shown in [68]. Table 1.2 is taken from Murota–Shioura [153].

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Chapter 2

Convex Functions with Combinatorial Structures

The objective of this chapter is to demonstrate how convex functions with combinatorial structures arise naturally from a variety of discrete systems, such as (i) discretizations of the Poisson equation, (ii) electrical networks consisting of linear (ohmic) and nonlinear resistors, and (iii) matrices (matroids) and polynomial matrices (valuated matroids). It is emphasized that such functions are always equipped with a pair of combinatorial properties, namely, submodularity (L-convexity) and exchangeability (M-convexity).

2.1

Quadratic Functions

In this section we see how quadratic convex functions with combinatorial structures arise naturally from linear discrete systems such as discretizations of the Poisson partial differential equations and electrical networks consisting of linear (ohmic) resistors. In so doing we intend to illustrate the rather vague idea of discreteness in direction introduced in the previous chapter. In accordance with the correspondence between quadratic functions and symmetric matrices, submodularity (L-convexity) and exchangeability (M-convexity) for quadratic functions and their conjugate functions are translated into combinatorial properties of symmetric matrices and their inverses.

2.1.1

Convex Quadratic Functions

A quadratic form is associated with a symmetric matrix A as27 f (x) =

1  x Ax. 2

(2.1)

Recall that a symmetric matrix A is said to be positive semidefinite if x Ax ≥ 0 for any vector x and positive definite if x Ax > 0 for any nonzero vector x. As is 27 The notation  means the transpose of a vector or a matrix. In section 2.1 we denote the ith component of a vector x by xi instead of x(i).

39

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Chapter 2. Convex Functions with Combinatorial Structures

well known, the convexity (resp., strict convexity) of f is equivalent to the positive semidefiniteness (resp., positive definiteness) of A: f is convex

⇔ A is positive semidefinite,

f is strictly convex ⇔ A is positive definite.

(2.2) (2.3)

Positive (semi)definiteness admits a number of characterizations. The first is in terms of eigenvalues: A is positive semidefinite ⇔ every eigenvalue of A is nonnegative, A is positive definite

⇔ every eigenvalue of A is positive.

(2.4) (2.5)

Note that the eigenvalues of a symmetric matrix are all real. The second characterization is in terms of minors (subdeterminants). Let N = {1, . . . , n} be the index set of rows and columns of A. For I ⊆ N and J ⊆ N we denote by A[I, J] the submatrix of A with row indices in I and column indices in J. A submatrix of the form A[I, I] for some I ⊆ N is called a principal submatrix and its determinant a principal minor . A principal submatrix of the form A[I, I] with I = {1, . . . , k} for some k (≤ n) is called a leading principal submatrix and its determinant a leading principal minor . Then we have A is positive semidefinite ⇔ every principal minor of A is nonnegative, (2.6) A is positive definite

⇔ every principal minor of A is positive,

(2.7)

and A is positive definite ⇔ every leading principal minor of A is positive.

(2.8)

The criterion (2.8) compares favorably with (2.7) in that there are only n leading principal minors as opposed to 2n principal minors. Positive (semi)definiteness can be checked with O(n3 ) arithmetic operations by an algorithm similar to Gaussian elimination. A change of the variable in (2.1), x = Sy with a nonsingular matrix S, results in another quadratic form fS (y) = f (Sy), which is associated with another symmetric matrix S  AS. The convexity of a quadratic form is preserved under such linear transformations of the variable, and the positive semidefiniteness of a symmetric matrix also remains invariant. The change of the variable x = Sy with a general nonsingular S, rotating the coordinate axes, does not respect any special coordinate directions. It would be reasonable to expect that such a general transformation should not be compatible with any combinatorial properties relevant to discreteness in direction. Conversely, we may regard properties of a quadratic form or of a symmetric matrix as being combinatorial with discreteness in direction if they are not invariant with respect to the entire class of transformations but are invariant with respect to some restricted subclass thereof (the class of diagonal scalings, for example). A typical combinatorial property of this kind is the sign pattern of the entries of a matrix. This is what we will study in the following subsection.

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2.1.2

41

Symmetric M-Matrices

As a typical combinatorial property we consider a particular sign pattern of a symmetric matrix that arises naturally in applications. The main theme here is the translation of this sign pattern of a symmetric matrix into a combinatorial property of the quadratic form associated with it. We consider symmetric matrices L = (ij | i, j = 1, . . . , n) that satisfy the following two conditions: ij ≤ 0 (i = j; 1 ≤ i, j ≤ n), n  [diagonal dominance] ij ≥ 0 (1 ≤ i ≤ n).

[off-diagonal nonpositivity]

(2.9) (2.10)

j=1

Note that the second condition (2.10) can also be expressed, under (2.9), as  ii ≥ |ij |. j =i

Such matrices often appear in applications, as demonstrated below. Example 2.1. Consider the Poisson equation −Δu = σ, where Δ is the Laplacian d 2 2 28 A i=1 d /dxi , σ denotes the source term, and d is the dimension of the space. standard discretization scheme for this differential equation, where we assume d = 1 for illustration purposes, gives rise to a system of linear equations described by a matrix like ⎡ ⎤ 2 −1 ⎢ −1 ⎥ 2 −1 ⎥. L=⎢ (2.11) ⎣ −1 2 −1 ⎦ −1 2 This matrix satisfies the two conditions (2.9) and (2.10) above.

Example 2.2. Consider the simple electrical network depicted in Fig. 2.1. It consists of five branches (linear resistors) connected at four nodes. We denote the conductance (the reciprocal of resistance) of branch j by gj > 0 (j = 1, . . . , 5), the potential at node i by pi (i = 1, . . . , 4), the voltage across branch j by ηj (j = 1, . . . , 5), and the current in branch j by ξj (j = 1, . . . , 5). The underlying graph can be represented by the incidence matrix ⎡ ⎤ −1 1 0 0 −1 ⎢ 1 0 0 −1 0 ⎥ ⎥, A=⎢ (2.12) ⎣ 0 −1 1 0 0 ⎦ 0 0 −1 1 1 whose rows and columns correspond, respectively, to the nodes and branches; the jth column has entry 1 at its initial node and −1 at its terminal node. The voltage 28 We

assume the Dirichlet boundary condition.

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Chapter 2. Convex Functions with Combinatorial Structures p1 I η1

η2 g1

 ξ1

g2 ξ2R

g5

p2

η5 ξ4 I

ξ5 6 g4

η4 R p 4

g3

?

p3 ξ3  η3

Figure 2.1. Electrical network . vector η = (ηj | j = 1, . . . , 5) is expressed in terms of the potential vector p = (pi | i = 1, . . . , 4) as η = A p. The constitutive equation (Ohm’s law) is represented as ξ = Y η, where ξ = (ξj | j = 1, . . . , 5) is the current vector and Y = diag (gj | j = 1, . . . , 5) is the conductance matrix. When a current source represented by a vector c = (ci | i = 1, . . . , 4) is applied, Kirchhoff’s current law is described by Aξ = c. Combining these equations yields AY A p = c for an admissible potential p. The coefficient matrix L = AY A here is given by ⎤ ⎡ −g1 −g2 −g5 g1 + g2 + g5 ⎥ ⎢ −g1 g1 + g4 0 −g4 ⎥, (2.13) L=⎢ ⎦ ⎣ −g2 0 g2 + g3 −g3 −g5 −g4 −g3 g3 + g4 + g5 which satisfies the two conditions (2.9) and (2.10) above. The matrix L is called the node admittance matrix . Note 2.3. A matrix L is called an M-matrix if it can be represented as L = sI − B with a matrix B consisting of nonnegative entries and a real number s ≥ ρ(B), where ρ(B) denotes the spectral radius (the largest modulus of an eigenvalue) of B. A nonsingular M-matrix is characterized as a matrix whose off-diagonal entries are all nonpositive and the entries of whose inverse matrix are all nonnegative. With this terminology we can say that symmetric matrices with off-diagonal nonpositivity and diagonal dominance considered in this section are exactly the same as diagonally dominant symmetric M-matrices. In passing we mention the fact that any symmetric M-matrix can be transformed into a diagonally dominant symmetric M-matrix by a symmetric diagonal scaling. M-matrices are fundamental concepts in ˇ control system theory (Kodama–Suda [113], Siljak [193]), numerical linear algebra (Axelsson [6], Varga [207]), and economics. The reader is referred to Berman– Plemmons [9] for mathematical properties of M-matrices. It is also mentioned that

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43

a symmetric compartmental matrix (Anderson [3]) is the same as the negative of a diagonally dominant symmetric M-matrix. Proposition 2.4. A symmetric matrix L with properties (2.9) and (2.10) is positive semidefinite. Proof. By (2.6) it suffices to show that any principal minor is nonnegative. We prove this by induction on the size n of the matrix L. Any principal submatrix of order ≤ n − 1 satisfies (2.9) and (2.10) and therefore its determinant is nonnegative by the induction hypothesis. It remains to show det L ≥ 0. Partition L as   L11 L12 , L= L21 L22 where L11 is of order n − 1 and L22 = nn . If nn = 0, then ni = in = 0 ˆ 11 = (i = 1, . . . , n − 1) and therefore det L = 0. Suppose nn > 0 and put L −1 −1 ˆ L11 − L12 L22 L21 . An off-diagonal entry ij − in nn nj of L11 , where 1 ≤ i = j ≤ n − 1, is nonpositive by ij , in , nj ≤ 0 and nn > 0. For the ith row sum of ˆ 11 we have L n−1 n−1 n  in  −in  ij − nj ≥ nj ≥ 0. nn j=1 nn j=1 j=1 ˆ 11 satisfies (2.9) and (2.10), and, by the induction hypothesis, its determiThus, L ˆ 11 ≥ 0. nant is nonnegative. Hence, det L = nn · det L We now look at the associated quadratic form 1  p Lp (p ∈ Rn ), (2.14) 2 which is convex by Proposition 2.4 and (2.2). Our goal here is to reveal a key combinatorial property of g(p) that reflects the combinatorial properties (2.9) and (2.10) of the matrix L. g(p) =

Note 2.5. It is quite natural to consider a quadratic form in association with a linear system of equations. For a positive-definite symmetric matrix L, the solution p to a system of linear equations Lp = c can be characterized as the unique minimizer of the quadratic function 12 p Lp − p c (variational formulation). Such a quadratic function often has a physical meaning. For instance, in the electrical network of Example 2.2, the function 1 1  p Lp − p c = η  ξ − p c (2.15) 2 2 represents the power (energy) consumed in the network. The Poisson equation −Δu = σ (with d = 1) in Example 2.1 can be translated into a variational problem of minimizing a functional   b 1  (u (x))2 − σ(x)u(x) dx. I[u] = 2 a

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Chapter 2. Convex Functions with Combinatorial Structures

In this case, the quadratic function represents a discretization of I[u]. For vectors p, q ∈ Rn , we denote the vectors of componentwise maxima and minima by p ∨ q and p ∧ q, respectively: (p ∨ q)i = max(pi , qi ),

(p ∧ q)i = min(pi , qi ).

(2.16)

A function g : Rn → R ∪ {+∞} is said to be submodular if it satisfies the inequality g(p) + g(q) ≥ g(p ∨ q) + g(p ∧ q)

(p, q ∈ Rn ).

(2.17)

This inequality is referred to as the submodularity inequality. It is understood that the inequality (2.17) is satisfied if g(p) = +∞ or g(q) = +∞. Submodularity corresponds to off-diagonal nonpositivity. Proposition 2.6. For a symmetric matrix L, the off-diagonal nonpositivity (2.9) of L is equivalent to the submodularity (2.17) of the associated quadratic form g(p). Proof. The inequality (2.17) for p = χi (ith unit vector) and q = χj (jth unit vector) yields (2.9). For the converse, put a = p ∧ q, p = a + pˆ, and q = a + qˆ. Then p ∨ q = a + pˆ + qˆ. Substitution of these into (2.14) shows that the right-hand side of (2.17) minus the left-hand side of (2.17) is represented as  q= pˆi ij qˆj ≤ 0, g(a + pˆ + qˆ) + g(a) − g(a + pˆ) − g(a + qˆ) = pˆ Lˆ i∈I j∈J

where I = {i | pˆi > 0} and J = {j | qˆj > 0}. Note that ij ≤ 0 (i ∈ I, j ∈ J) by I ∩ J = ∅. Off-diagonal nonpositivity can thus be translated into submodularity. Then how does the combination of the off-diagonal nonpositivity and the diagonal dominance of L translate to g(p)? To this end, we strengthen submodularity to (SBF [R]) g(p) + g(q) ≥ g((p − α1) ∨ q) + g(p ∧ (q + α1)) (∀ p, q ∈ Rn , ∀ α ∈ R+ ), which we call translation submodularity. Submodularity (2.17) is a special case of this with α = 0. Theorem 2.7. For a symmetric matrix L, conditions (a) and (b) are equivalent. (a) L has off-diagonal nonpositivity (2.9) and diagonal dominance (2.10). (b) g(p) has translation submodularity (SBF [R]). Proof. (b) ⇒ (a): Proposition 2.6 shows (2.9). (SBF [R]) with p = χi , q = −1, and α = 1 yields (2.10), since  2[g(p) + g(q)] = χ i Lχi + 1 L1,   2[g((p − α1) ∨ q) + g(p ∧ (q + α1))] = χ i Lχi + 1 L1 − 2χi L1.

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45

(a) ⇒ (b): Put I = {i | α ≤ pi − qi }, and let J be the complement of I. We have   pi − α (i ∈ I), qi + α (i ∈ I), ((p − α1) ∨ q)i = (p ∧ (q + α1))i = qi pi (i ∈ J), (i ∈ J). With the use of k∈I ik ≥ − j∈J ij , which is a consequence of (2.10), as well as (2.9), we obtain [g(p) + g(q)] − [g((p − α1) ∨ q) + g(p ∧ (q + α1))]    = α(pi − qi − α) ik + (pi − qi − α)(pj − qj )ij i∈I

≥−



k∈I

α(pi − qi − α)

i∈I

=



 j∈J

i∈I j∈J

ij +



(pi − qi − α)(pj − qj )ij

i∈I j∈J

(pi − qi − α)(pj − qj − α)ij ≥ 0.

i∈I j∈J

Hence follows (SBF [R]). Theorem 2.7 and Proposition 2.4 show that translation submodularity =⇒ convexity for a quadratic function. The converse, however, is not true. It is emphasized that translation submodularity is a combinatorial property in that it is not invariant under coordinate rotations but respects the fixed coordinate axes and the particular direction 1. Note 2.8. The quadratic form considered in this section coincides with what is known as the Dirichlet form (in finite dimension) in the theory of the Markov process and potential theory; see Fukushima–Oshima–Takeda [71]. We mention here the equivalence among the five conditions (a), (b), (c), (d), and (e) below.29 The five conditions are equivalent, by Theorem 2.7, to the translation submodularity (SBF [R]) of g(p). (a) L has off-diagonal nonpositivity (2.9) and diagonal dominance (2.10). (b) The normal contraction operates on g(p) = 12 p Lp; i.e., for p, q ∈ Rn , |pi | ≥ |qi |, |pi − pj | ≥ |qi − qj | (1 ≤ i, j ≤ n) =⇒ g(p) ≥ g(q). (c) Every unit contraction operates on g(p) =

1 2

p Lp; i.e.,

g(p) ≥ g((0 ∨ p) ∧ 1) (p ∈ Rn ).  −1 exists and is Markovian; i.e., (d) For any α > 0, Sα = I + α1 L 0 ≤ x ≤ 1 =⇒ 0 ≤ Sα x ≤ 1. 29 In

the terminology of the theory of Dirichlet forms, −L corresponds to the generator , α−1 Sα to the resolvent, and Tt to the semigroup.

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Chapter 2. Convex Functions with Combinatorial Structures

(e) For any t > 0, Tt = exp(−tL) is Markovian. The proof of the equivalence of the above five conditions follows. [(a) ⇒ (b)]: Denote the distinct values in {pi | 1 ≤ i ≤ n} ∪ {0} by π1 > π2 > · · · > πm , and put X = {i | pi = π1 } and Y = {i | pi = πm }. It suffices to prove g(p) ≥ g(q) for q = p − αχX with 0 ≤ α ≤ 2(π1 − π2 ) or q = p + βχY with 0 ≤ β ≤ 2(πm−1 − πm ), since any normal contraction q can be obtained from p by a series of transformations of such forms. We consider the former case (the latter can be dealt with similarly). We may assume X = ∅, α > 0, and π1 > π2 ≥ 0. Then we have 1 (g(p) − g(q)) α n !   α α"   = ij pj − ij = π1 − ij + ij pj 2 2 j=1 i∈X

≥ π2

 i∈X j∈X

i∈X j∈X

ij +



i∈X j∈X

ij π2 = π2

i∈X j ∈X /

n 

i∈X j ∈X /

ij ≥ 0.

i∈X j=1

[(b) ⇒ (c)]: Note that q = (0 ∨ p) ∧ 1 is a normal contraction of p. [(c) ⇒ (a)]: Let α be a sufficiently small positive number. Then (2.9) follows from (c) for p = χi − αχj and (2.10) from (c) for p = 1 + αχi . [(c) ⇒ (d)]: Since (c) ⇒ (a), L is positive semidefinite by Proposition 2.4, and therefore, Sα exists. For a fixed x with 0 ≤ x ≤ 1 and α > 0, the function α ψ(p) = g(p) + (p − x) (p − x) 2 takes the unique minimum at p = p0 , where p0 = Sα x. For q0 = (0 ∨ p0 ) ∧ 1 we have g(p0 ) ≥ g(q0 ) by (c) and (p0 − x) (p0 − x) ≥ (q0 − x) (q0 − x) by 0 ≤ x ≤ 1. Hence ψ(p0 ) ≥ ψ(q0 ), which implies p0 = q0 = (0 ∨ p0 ) ∧ 1. This shows 0 ≤ p0 = Sα x ≤ 1. n [(d) ⇒ (c)]: Since Sα = (sij ) is Markovian, we have sij ≥ 0 (1 ≤ i, j ≤ n) and j=1 sij ≤ 1 (1 ≤ i ≤ n). Define g

(α)

−1  1  1 (p) = p Lp I+ L 2 α

for α > 0. Then g (α) (p) tends to g(p) as α → +∞. On the other hand, the expression ⎛ ⎞ n  n n n    α ⎝1 − sij (pi − pj )2 + α sij ⎠ p2i 2g (α) (p) = α(p p − p Sα p) = 2 i=1 j=1 i=1 j=1 shows that every normal contraction operates on g (α) (p). The limit of α → +∞ establishes (c). [(d) ⇒ (e)]: This is due to the formula Tt x = lim e−αt α→+∞

 (αt)n (Sα )n x. n!

n≥0

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47

[(e) ⇒ (d)]: This is due to the formula Sα x = α

2.1.3

'∞ 0

e−αt Tt x dt.

Combinatorial Property of Conjugate Functions

As a continuation of our study of quadratic convex functions with translation submodularity, we now consider the conjugate of such functions. The conjugate of a quadratic form is another quadratic form, which is associated with the matrix inverse of the original coefficient matrix. Proposition 2.9. Let M and L be positive-definite symmetric matrices. The quadratic forms f (x) = 12 x M x and g(p) = 12 p Lp are conjugate to each other with respect to the Legendre–Fenchel transformation (1.6) if and only if M and L are inverse to each other. Proof. This follows from a straightforward calculation based on (1.6). Hence, part of our study consists of investigating the combinatorial structure of the inverse of symmetric matrices with off-diagonal nonpositivity (2.9) and diagonal dominance (2.10). We introduce the following notation: L = {L | L is positive definite and satisfies (2.9) and (2.10)}, L−1 = {L−1 | L ∈ L}. Example 2.10. Recall the Poisson equation of the matrix L in (2.11) is given by ⎡ 4 3 1⎢ 3 6 M= ⎢ 5⎣ 2 4 1 2

(2.18) (2.19)

−Δu = σ in Example 2.1. The inverse 2 4 6 3

⎤ 1 2 ⎥ ⎥. 3 ⎦ 4

(2.20)

Whereas the matrix L represents a differential operator, M = L−1 corresponds to the Green function. The function g(p) is an approximation to the functional I[u] for the variational formulation (Note 2.5), and the conjugate of g(p) is that for the inverse problem of finding σ for a given u. Let us consider the quadratic form f (x) =

1  x Mx 2

associated with M ∈ L−1 . We are to show that f (x) possesses an exchange property: (M -EXC[R]) For x, y ∈ domR f and i ∈ supp+ (x − y), there exist j ∈ supp− (x − y) ∪ {0} and a positive number α0 ∈ R++ such that f (x) + f (y) ≥ f (x − α(χi − χj )) + f (y + α(χi − χj )) for all α ∈ R with 0 ≤ α ≤ α0 .

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It should be clear that χi designates the ith unit vector for 1 ≤ i ≤ n while χ0 is the zero vector and supp+ (x) = {i | xi > 0},

supp− (x) = {i | xi < 0}

(2.21)

for x = (xi | i = 1, . . . , n) ∈ Rn . Recall that such an exchange property can be viewed as a combinatorial analogue of the basic inequality (1.39) valid for a general convex function. The following is the main theorem of this section, stating the conjugacy relationship between L -convexity and M -convexity for strictly convex quadratic functions. Theorem 2.11. Suppose that strictly convex quadratic forms g and f are conjugate to each other. Then g satisfies translation submodularity (SBF [R]) if and only if f has exchange property (M -EXC[R]). The conjugacy relationship between (SBF [R]) and (M -EXC[R]) stated above for strictly convex quadratic forms is in fact valid for a more general class of functions, as is fully developed in Chapter 8. This particular case, however, deserves separate consideration, in that it admits a matrix-algebraic proof using the Farkas lemma and thereby provides a new insight into (SBF [R]) vs. (M -EXC[R]) conjugacy. In what follows we prove Theorem 2.11 by establishing Theorem 2.12 below.30 As variants of (M -EXC[R]) we consider the following: (M -EXC+ [R]) For x, y ∈ domR f and i ∈ supp+ (x − y), there exist j ∈ supp− (x − y) ∪ {0} and a positive number α0 ∈ R++ such that f (x) + f (y) > f (x − α(χi − χj )) + f (y + α(χi − χj )) for all α ∈ R with 0 < α < α0 . (M -EXCd [R]) For x, y ∈ domR f and i ∈ supp+ (x − y), min

[f  (x; −χi + χj ) + f  (y; χi − χj )] ≤ 0.

j∈supp− (x−y)∪{0}

The latter is motivated by the identity f (x + αd) = f (x) + αf  (x; d) + O(α2 )

(2.22)

valid for the directional derivative f  (x; d) and sufficiently small α > 0. We also  denote by (M -EXC+ d [R]) the property (M -EXCd [R]) with ≤ replaced by strict inequality min 0, min x mj . j =i

(c) For any x ∈ Rn and i ∈ supp+ (x),  x mi ≥ min 0, 

min

j∈supp− (x)





x mj .

(c+ ) For any x ∈ Rn and i ∈ supp+ (x),  x mi > min 0, 

(d) f (x) =

1 2

(d+ ) f (x) = (e) f (x) =

1 2

(e+ ) f (x) =

min

j∈supp− (x)





x mj .

x M x satisfies (M -EXCd [R]). 1 2

x M x satisfies (M -EXC+ d [R]).

x M x satisfies (M -EXC[R]). 1 2

x M x satisfies (M -EXC+ [R]).

Proof. We prove the equivalence by showing the following implications: (a) !

=⇒ (b+ ) ← ↓⇑ (b) ←

(c+ ) ⇔ ↓ (c) ⇔

(d+ ) ⇒ (e+ ) ↓ ↓ (d) ← (e)

The implications indicated by ← or ↓ are easy to see and those by ⇔ or ⇑ are proved below. n (a) ⇔ (b+ ): Denoting M −1 by L = (ij ), we have M L = I; i.e., j=1 ji mj = χi , which can be rewritten as ⎛ ⎞ n   ⎝ ji ⎠ mi + (−ji )(mi − mj ) = χi . j=1

j =i

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Chapter 2. Convex Functions with Combinatorial Structures

The condition (a), L ∈ L, is equivalent to all the coefficients in this expression being nonnegative, whereas the latter is equivalent, by the Farkas lemma (Proposition 2.13 below), to   x χi > 0 ⇒ max x mi , max x (mi − mj ) > 0. j =i

(2.23)

This is nothing but (b+ ), since x χi > 0 is the same as i ∈ supp+ (x). (b) ⇒ (b+ ): This follows from the above argument and the latter half of Proposition 2.13 below. (a) ⇒ (c+ ): Fix x ∈ Rn . It suffices to show     min x mi > min 0, min x mj . i∈supp+ (x)

j∈supp− (x)

Let i ∈ supp+ (x) attain the minimum on the left-hand side. Put S = supp+ (x) ∪ supp− (x) and let x ∈ RS denote the restriction of x to S. The submatrix of M with row and column indices in S is denoted by M = (mj | j ∈ S), where mj ∈ RS . Then we have supp+ (x) = supp+ (x), supp− (x) = supp− (x), i ∈ supp+ (x), xj = 0 (∀ j ∈ S), and x mj = x mj (∀ j ∈ S). Since M ∈ L−1 by Proposition 2.14 below, M satisfies (b+ ). Hence         x mi > min 0, min x mj = min 0, x mj , min x mj , min j∈supp+ (x)\{i}

j =i

j∈supp− (x)

in which x mi ≤

min

j∈supp+ (x)\{i}

x mj

by the choice of i. Hence, we obtain  x mi > min 0,

min

j∈supp− (x)

 x  mj .

(c+ ) ⇔ (d+ ), (c) ⇔ (d): Using f  (x; d) = x M d we obtain f  (x; −χi ) + f  (y; χi ) = −x M χi + y  M χi = −(x − y) mi as well as a similar expression for f  (x; −χi + χj ) + f  (y; χi − χj ). We then replace x − y with x. (d+ ) ⇒ (e+ ): This follows easily from (2.22).

Proposition 2.13. For a matrix A and a vector b, the conditions (a) and (b) below are equivalent (Farkas lemma): 31 (a) Ax = b for some nonnegative x ≥ 0. 31 Inequality between vectors means componentwise inequality; e.g., x ≥ 0 for x = (x )n i i=1 means xi ≥ 0 for i = 1, . . . , n.

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(b) y  b ≥ 0 for any y such that y  A ≥ 0 . If A is nonsingular, condition (b) is equivalent to (c) y  b ≥ 0 for any y such that y  A > 0 . Proof. (a) ⇒ (b) ⇒ (c) is obvious. (b) ⇒ (a) is proved later in Theorem 3.9. For (c) ⇒ (b), there exists z such that z  A = 1 by the assumed nonsingularity of A. Then (y + εz) A > 0 for any ε > 0, and (c) yields (y + εz) b ≥ 0, which implies y  b ≥ 0, since ε > 0 is arbitrary. Proposition 2.14. Any principal submatrix of M ∈ L−1 belongs to L−1 . Proof. Partition M and L = M −1 compatibly as    M11 M12 L11 M= , L= M21 M22 L21

L12 L22

 .

To prove M11 ∈ L−1 by induction on the size of M11 , we may assume M22 and L22 ˆ 11 ), are 1 × 1. Since L22 = nn > 0, we have M11 −1 = L11 − L12 L22 −1 L21 (= L which shows the nonsingularity of M11 . Then the proof of Proposition 2.4 shows M11 ∈ L−1 . Note 2.15. It is worth noting that conditions (c) and (c+ ) in Theorem 2.12 immediately imply positive semidefiniteness and positive definiteness, respectively. The proof for the former reads as follows, while a similar proof works for the latter. Let μ be an eigenvalue of the matrix M and x be the corresponding eigenvector with supp+ (x) = ∅. Then (c) shows   μxi ≥ min 0, min− μxj j∈supp (x)

for i ∈ supp+ (x). This implies μ ≥ 0, since, otherwise, the left-hand side is negative and the right-hand side is zero. Hence M is positive semidefinite by (2.4). It is also noted that mii ≥ 0 follows from (c) with x = χi and mij ≥ 0 from (c) with x = χi + αχj with α > 0 large.

2.1.4

General Quadratic L-/M-Convex Functions

We have so far considered strictly convex quadratic forms defined by positivedefinite matrices. The conjugacy relationship carries over to convex quadratic forms defined by positive-semidefinite matrices, as follows. Suppose that g is a quadratic convex function given by ( 1 p Lp (p ∈ K), g(p) = 2 +∞ (p ∈ / K)

(2.24)

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Chapter 2. Convex Functions with Combinatorial Structures

with a positive-semidefinite symmetric matrix L and a linear subspace K ⊆ Rn . Then the conjugate of g is also a quadratic convex function given by ( 1 x M x (x ∈ H), (2.25) f (x) = 2 +∞ (x ∈ / H) with a positive-semidefinite symmetric matrix M and a linear subspace H ⊆ Rn such that H = (K ∩ ker L)⊥ , (2.26) K = (H ∩ ker M )⊥ , where ker M = {x ∈ Rn | M x = 0}, X ⊥ = {p ∈ Rn | p, x = 0 (∀ x ∈ X)}. Note that (2.26) can be rewritten as domR g = (arg min f )⊥ ,

domR f = (arg min g)⊥ .

The conjugacy stated in Theorem 2.11 for strictly convex quadratic forms is generalized as follows. See Murota–Shioura [155] for the proof as well as the structure of the coefficient matrices L and M . Theorem 2.16. Suppose that g : Rn → R ∪ {+∞} in (2.24) and f : Rn → R ∪ {+∞} in (2.25) are conjugate to each other. Then g satisfies translation submodularity (SBF [R]) if and only if f has exchange property (M -EXC[R]).

2.2

Nonlinear Networks

In the previous section we have seen that an electrical network gives rise to a convex function with combinatorial properties (Example 2.2). The node admittance matrix L is a diagonally dominant symmetric M-matrix (with off-diagonal nonpositivity and diagonal dominance), and the associated quadratic function (2.15) representing the power (energy) consumed in the network has translation submodularity (SBF [R]). In this section we shall see a similar phenomenon in an electrical network of nonlinear resistors or, equivalently, in a nonlinear minimum cost flow problem. General convex functions, not necessarily quadratic, arise as a result of nonlinearity. Two aspects of discreteness, discreteness in direction and discreteness in value, both appear naturally in the network flow problem. Accordingly, we consider functions of type Rn → R in section 2.2.1 and those of type Zn → Z in section 2.2.2.

2.2.1

Real-Valued Flows

Let G = (V, A) be a directed graph with the set of vertices (nodes) V and the set of arcs (branches) A and T be a set of distinguished vertices called terminals; see Fig. 2.2. For each vertex v ∈ V , δ + v and δ − v denote the sets of arcs leaving v and

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T

f (x) g(p)

53 (fa , ga ) j *

)  s q

j

N

W

O

1 (ξ(a), η(a))

Figure 2.2. Multiterminal network . entering v, respectively. For each arc a ∈ A, ∂ + a designates the initial vertex of a and ∂ − a the terminal vertex of a. We consider here a minimum cost flow problem, in which each arc is associated with a nonlinear convex cost function and the supply (or demand) of flow is specified at terminal vertices. The physical model we have in mind is a multiterminal electrical network that consists of nonlinear resistors and is driven by a (current or voltage) source applied to the terminal vertices. To reinforce physical intuition, we sometimes use terminology such as current and voltage instead of flow and tension, but no physics is really involved in our arguments. A reader who is more comfortable with combinatorial optimization terminology may replace the terminology as follows: electrical network current voltage current source potential current potential voltage potential characteristic curve

⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒

network flow tension supply of flow potential (dual variable) cost function in flow cost function in tension kilter diagram

Each arc a ∈ A is associated with a flow (or current ) ξ(a) and a tension (or voltage) η(a) and each vertex v ∈ V with a potential p˜(v). The boundary of flow ξ is defined to be32   ∂ξ(v) = {ξ(a) | a ∈ δ + v} − {ξ(a) | a ∈ δ − v} (v ∈ V ), (2.27) which represents the net flow leaving vertex v. The coboundary of potential p˜ is δ p˜(a) = p˜(∂ + a) − p˜(∂ − a)

(a ∈ A),

(2.28)

which expresses the difference in the potentials at the end vertices of an arc a. 32 Rockafellar’s

notations div and Δ in [178] are related to ours by div = ∂ and Δ = −δ.

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Chapter 2. Convex Functions with Combinatorial Structures η

(ξ, η)

6

Γa ga (η) fa (ξ)

ξ

Figure 2.3. Characteristic curve.

For each terminal vertex v ∈ T , let x(v) denote the amount of flow going out of the network at v and p(v) be the potential at v. We have structural equations  −x(v) (v ∈ T ), ∂ξ(v) = η(a) = −δ p˜(a) (a ∈ A), (2.29) 0 (v ∈ V \ T ), expressing the conservation laws as well as an obvious relation p(v) = p˜(v)

(v ∈ T ).

(2.30)

The vectors x = (x(v) | v ∈ T ) ∈ RT and p = (p(v) | v ∈ T ) ∈ RT play primal roles in our discussion below. Each arc a ∈ A is associated with a characteristic curve Γa ⊆ R2 , which describes the admissible pairs of flow ξ(a) and tension η(a): (ξ(a), η(a)) ∈ Γa

(a ∈ A).

(2.31)

In physical terms a characteristic curve shows the constitutive equation for a nonlinear resistor, describing the possible pairs of current and voltage. In the linear case (as in Example 2.2) we have Γa = {(ξ, η) ∈ R2 | η = Ra ξ} for some Ra > 0, which represents the resistance of an ohmic resistor. We consider here a nonlinear case, where monotonicity (ξ1 , η1 ), (ξ2 , η2 ) ∈ Γa =⇒ (ξ1 − ξ2 ) · (η1 − η2 ) ≥ 0

(2.32)

is assumed (see Fig. 2.3). A conjugate pair of convex functions are induced from the characteristic curve Γa . Define  (ξ,η)  (ξ,η) fa (ξ) = ηdξ, ga (η) = ξdη, (2.33) Γa

Γa

which means that fa (ξ) is the area below Γa in Fig. 2.3 and ga (η) is the area above Γa . The functions fa (ξ) and ga (η) are both convex as a consequence of the assumed

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55

monotonicity (2.32). Moreover, with suitable choices of integral constants, we have fa (ξ) + ga (η) ≥ ξη

(∀ (ξ, η) ∈ R2 ),

fa (ξ) + ga (η) = ξη ⇐⇒ (ξ, η) ∈ Γa .

(2.34) (2.35)

Hence follows fa (ξ) = sup{ξη − ga (η) | η ∈ R},

ga (η) = sup{ξη − fa (ξ) | ξ ∈ R}.

(2.36)

That is, fa and ga are conjugate to each other with respect to the Legendre–Fenchel transformation (1.6). In the theory of electrical networks, the function fa (ξ) is sometimes called the current potential (or content ) and ga (η) the voltage potential (or cocontent ). In the case of a linear resistor, the functions fa and ga are quadratic, i.e., fa (ξ) =

Ra 2 ξ , 2

ga (η) =

1 2 η , 2Ra

and they are both equal to half the power consumed in the resistor. When a current source described by x ∈ RT with v∈T x(v) = 0 is applied to the terminal vertices of the network, the equilibrium state of (ξ(a) | a ∈ A) and (η(a) | a ∈ A) is determined as a solution to the structural equations (2.29) and the constitutive equations (2.31). A variational formulation of this problem is to minimize the total current potential a∈A fa (ξ(a)) among all possible current distributions ξ subject to the conservation law (2.29), with the current vector at the equilibrium state being characterized as a minimizer of this problem (see Note 2.18 in section 2.2.3). We define f (x) to be the minimum value of the total current potential in this variational problem; i.e., ) * ( )  ) fa (ξ(a))) ∂ξ(v) = −x(v) (v ∈ T ), ∂ξ(v) = 0 (v ∈ V \ T ) . (2.37) f (x) = inf ) ξ a∈A

When a voltage source described by p ∈ RT (with respect to some reference point) is applied to the terminal vertices of the network, the equilibrium state of (ξ(a) | a ∈ A) and (η(a) | a ∈ A) is determined as a solution to the structural equations (2.29) and (2.30) and the constitutive equations (2.31). A variational formulation of this problem is to minimize the total voltage potential a∈A ga (η(a)) among all possible voltage distributions η subject to the conservation law (2.29) and (2.30), with the voltage vector at the equilibrium state being characterized as a minimizer of this problem (see Note 2.18). We define g(p) to be the minimum value of the total voltage potential in this variational problem; i.e., ) * ( )  ) (2.38) ga (η(a))) η(a) = −δ p˜(a) (a ∈ A), p˜(v) = p(v) (v ∈ T ) . g(p) = inf η,p˜ ) a∈A

The functions f and g introduced above are both convex (see Note 2.17 in section 2.2.3) and they are conjugate to each other (see Note 2.18). In this sense

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they stand on equal footing and there seems to be no concept in convex analysis that distinguishes between f and g. When it comes to combinatorial properties, however, these functions have distinctive features. As is shown in Notes 2.19 and 2.20 in section 2.2.3, the function f : RT → R ∪ {+∞} is endowed with an exchange property: (M-EXC[R]) For x, y ∈ domR f and u ∈ supp+ (x − y), there exist v ∈ supp− (x − y) and a positive number α0 ∈ R++ such that f (x) + f (y) ≥ f (x − α(χu − χv )) + f (y + α(χu − χv )) for all α ∈ R with 0 ≤ α ≤ α0 , and the function g : RT → R ∪ {+∞} satisfies (SBF[R]) g(p) + g(q) ≥ g(p ∨ q) + g(p ∧ q) (∀ p, q ∈ RT ), (TRF[R]) ∃ r ∈ R such that g(p + α1) = g(p) + αr (∀ p ∈ RT , ∀ α ∈ R) with r = 0. (SBF[R]) is the submodularity and (TRF[R]) with r = 0 is the invariance in the direction of 1 = (1, 1, . . . , 1), which corresponds to the fact that the reference point of a potential can be chosen arbitrarily. Recall that we have already seen (M-EXC[R]) in the definition of M-convex functions in section 1.4.2 and (SBF[R]) and (TRF[R]) in the definition of L-convex functions in section 1.4.1. The following points are emphasized here for the conjugate pair of convex functions, f and g, appearing in the network flow problem: • The functions f and g cannot be categorized with respect to convexity alone. • The functions f and g can be classified into different categories (M-convexity and L-convexity) with respect to combinatorial properties. • Exchangeability (M-convexity) and submodularity (L-convexity) appear as conjugate properties. The combinatorial properties (M-EXC[R]), (SBF[R]), and (TRF[R]) above capture the kind of discreteness we call discreteness in direction. In the next subsection we turn to the other kind of discreteness, discreteness in value, inherent in the network flow problem, by considering integer-valued flows.

2.2.2

Integer-Valued Flows

By replacing R in the previous argument with Z systematically, we consider integervalued flows in a network specified by integral data. In particular, we assume all the vectors representing flow, tension, potential, etc., are integer valued; i.e., ξ ∈ ZA , η ∈ ZA , p˜ ∈ ZV , x ∈ ZT , p ∈ ZT , etc. As for the cost functions, we assume that each arc a ∈ A is associated with a pair of integer-valued functions fa , ga : Z → Z ∪ {+∞} such that fa (ξ − 1) + fa (ξ + 1) ≥ 2fa (ξ) ga (η − 1) + ga (η + 1) ≥ 2ga (η)

(∀ ξ ∈ Z), (∀ η ∈ Z),

(2.39) (2.40)

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K

57

fa 6 8 7 6 5 4 3 2 1

6

6

−3 −2 −1 0 1 2 3 ξ

ga 6 8 7 6 5 4 3 2 1



−2 −1 0 1 2 3 4 5 η

Figure 2.4. Conjugate discrete convex functions fa (ξ) and ga (η).

η 6 6 5 4 3 2 1 −4 −3 −2 

0

6

1 2 3

ξ

−2

Figure 2.5. Discrete characteristic curve Γa .

and fa (ξ) = sup{ξη − ga (η) | η ∈ Z},

ga (η) = sup{ξη − fa (ξ) | ξ ∈ Z}.

(2.41)

It should be clear that (2.41) is a discrete analogue of the conjugacy (2.36), the discrete Legendre–Fenchel transformation (1.9) for univariate functions. An example of such a conjugate pair of cost functions is demonstrated in Fig. 2.4. The characteristic curve Γa in this discrete setting is defined to be a subset of Z2 induced from (fa , ga ) by (2.35). It can be characterized as a subset of Z2 with the monotonicity property (2.32). Figure 2.5 shows the characteristic curve Γa induced from (fa , ga ) in Fig. 2.4. In parallel with (2.37) and (2.38) we define functions f : ZT → Z ∪ {+∞} and

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Chapter 2. Convex Functions with Combinatorial Structures

g : ZT → Z ∪ {+∞} by ) ( * )  ) f (x) = inf fa (ξ(a))) ∂ξ(v) = −x(v)(v ∈ T ), ∂ξ(v) = 0(v ∈ V \ T ) , (2.42) ξ ) a∈A ) * ( )  ) ga (η(a))) η(a) = −δ p˜(a) (a ∈ A), p˜(v) = p(v)(v ∈ T ) . (2.43) g(p) = inf η,p˜ ) a∈A

Note that these expressions are identical to (2.37) and (2.38) except that the vectors are now integer valued and, in particular, the infima are taken over integer vectors. Fortunately, such a discretization in the definitions of f and g does not destroy the combinatorial properties discussed above. On the contrary, the discretization turns out to be compatible with natural discretizations of the combinatorial properties. Namely, it can be shown (see Note 2.19) that the function f has a discrete version of the exchange property: (M-EXC[Z]) For x, y ∈ domZ f and u ∈ supp+ (x − y), there exists v ∈ supp− (x − y) such that f (x) + f (y) ≥ f (x − χu + χv ) + f (y + χu − χv ). This is essentially the same as (M-EXC[R]) with α0 = α = 1. On the other hand, the function g satisfies (SBF[Z]) (TRF[Z])

g(p) + g(q) ≥ g(p ∨ q) + g(p ∧ q) (∀ p, q ∈ ZT ), ∃ r ∈ Z such that g(p + 1) = g(p) + r (∀ p ∈ ZT )

with r = 0 (see Note 2.20). Furthermore, these functions f and g are conjugate to each other with respect to the discrete Legendre–Fenchel transformation (1.9), as is proved later in section 9.6. We have thus seen that the transition from R to Z is quite smooth in the network flow problem. The combinatorial properties are discretized compatibly in the discretization of the problem data. We emphasize that this is by no means a general phenomenon but is an outstanding characteristic of the network flow problem.

2.2.3

Technical Supplements

This section provides a series of proofs for the major properties of the functions ) ( * )  ) f (x) = inf fa (ξ(a))) ∂ξ(v) = −x(v) (v ∈ T ), ∂ξ(v) = 0 (v ∈ V \ T ) , (2.44) ξ ) a∈A ) * ( )  ) (2.45) ga (η(a))) η(a) = −δ p˜(a) (a ∈ A), p˜(v) = p(v) (v ∈ T ) g(p) = inf η,p˜ ) a∈A

defined in (2.37) and (2.38).

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Note 2.17. We prove that f in (2.44) and g in (2.45) are convex functions under the assumption that f > −∞ and g > −∞. To show the convexity of f , fix x, y ∈ domR f . For any ε > 0 there exist ξx and ξy such that   f (x) + ε ≥ fa (ξx (a)), f (y) + ε ≥ fa (ξy (a)), (2.46) a∈A

a∈A

(∂ξx )|T = −x, (∂ξy )|T = −y, (∂ξx )|V \T = (∂ξy )|V \T = 0,

(2.47)

where · |T denotes the restriction of a vector to T . For λ ∈ [0, 1]R we have ∂(λξx + (1 − λ)ξy ) = λ∂ξx + (1 − λ)∂ξy and, therefore,  [λfa (ξx (a)) + (1 − λ)fa (ξy (a))] λf (x) + (1 − λ)f (y) + ε ≥ ≥



a∈A

fa (λξx (a) + (1 − λ)ξy (a)) ≥ f (λx + (1 − λ)y).

a∈A

This implies λf (x) + (1 − λ)f (y) ≥ f (λx + (1 − λ)y), since ε > 0 is arbitrary. To show the convexity of g, fix p, q ∈ domR g. For any ε > 0 there exist ηp , ηq , p˜, and q˜ such that   ga (ηp (a)), g(q) + ε ≥ ga (ηq (a)), (2.48) g(p) + ε ≥ a∈A

ηp = −δ p˜,

a∈A

ηq = −δ q˜,

p˜|T = p,

q˜|T = q.

(2.49)

For λ ∈ [0, 1]R we have δ(λ˜ p + (1 − λ)˜ q ) = λδ p˜ + (1 − λ)δ q˜ = −[ληp + (1 − λ)ηq ] and, therefore,  [λga (ηp (a)) + (1 − λ)ga (ηq (a))] λg(p) + (1 − λ)g(q) + ε ≥ ≥



a∈A

ga (ληp (a) + (1 − λ)ηq (a)) ≥ g(λp + (1 − λ)q).

a∈A

This implies λg(p) + (1 − λ)g(q) ≥ g(λp + (1 − λ)q), since ε > 0 is arbitrary. Note 2.18. The variational formulations of the network equilibrium are derived here under the assumption of the existence of an equilibrium. Also shown is the conjugacy between f in (2.44) and g in (2.45). It follows from (2.34) that  [fa (ξ(a)) + ga (η(a))] ≥ η, ξ = − δ p˜, ξ = − ˜ p, ∂ξ = p, x

(2.50) a∈A

for any ξ, η, p˜, x, and p satisfying the conservation laws expressed by (2.29) and (2.30). By (2.35), the inequality above is an equality if and only if (ξ(a), η(a)) ∈ Γa for each a ∈ A. Suppose that an equilibrium state exists when a current source x = x∗ is applied to the network and let ξ ∗ , η ∗ , p˜∗ , p∗ be the vectors at the equilibrium. The inequality (2.50) with η = η ∗ , p˜ = p˜∗ , p = p∗ , and x = x∗ yields   fa (ξ(a)) ≥ p∗ , x∗ − ga (η ∗ (a)) for all ξ satisfying (2.29), a∈A

a∈A

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Chapter 2. Convex Functions with Combinatorial Structures

which shows that the minimum of the current potential a∈A fa (ξ(a)) is attained by ξ = ξ ∗ . A similar result occurs for the variational formulation using the voltage potential a∈A ga (η(a)) when a voltage source described by p = p∗ is applied. For the conjugacy of f and g, note that ( ( * *   f (x) + g(p) = inf fa (ξ(a)) + inf ga (η(a)) ≥ p, x

ξ

a∈A

η

a∈A

follows from (2.50) and the definitions of f and g in (2.44) and (2.45). This inequality for x = x∗ turns into an equality for p = p∗ , which shows that f (x∗ ) = supp { p, x∗ − g(p)} = g • (x∗ ). A similar argument works for g = f • . The argument here is admittedly lacking in mathematical rigor, for which the reader is referred to Rockafellar [178]. Note 2.19. We prove that f in (2.44) satisfies (M-EXC[R]) under the assumption that f > −∞. Fix x, y ∈ domR f . For any ε > 0 there exist ξx and ξy satisfying (2.46) and (2.47). Consider the difference in the flows, ξy − ξx ∈ RA , for which we have  x(v) − y(v) (v ∈ T ), ∂(ξy − ξx )(v) = 0 (v ∈ V \ T ). Since u ∈ supp+ (x − y), there exists a path compatible with ξy − ξx that connects u to some vertex in supp− (x − y) (i.e., an augmenting path with respect to the pair of flows ξx and ξy ). More formally, there exist π : A → {0, ±1} and v ∈ supp− (x − y) such that ˜u − χ ˜v , supp+ (π) ⊆ supp+ (ξy − ξx ), supp− (π) ⊆ supp− (ξy − ξx ), ∂π = χ where χ ˜u , χ ˜v ∈ RV are the characteristic vectors of u and v. For two flows ξx + απ and ξy − απ with 0 ≤ α ≤ α0 , where α0 = min |ξy (a) − ξx (a)| (> 0), we have a:|π(a)|=1

ξx (a) > ξy (a) ⇒ fa (ξx (a) − α) + fa (ξy (a) + α) ≤ fa (ξx (a)) + fa (ξy (a)), ξx (a) < ξy (a) ⇒ fa (ξx (a) + α) + fa (ξy (a) − α) ≤ fa (ξx (a)) + fa (ξy (a)). It then follows that f (x − α(χu − χv )) + f (y + α(χu − χv ))  ≤ [fa (ξx (a) + απ(a)) + fa (ξy (a) − απ(a))] a∈A





[fa (ξx (a)) + fa (ξy (a))] ≤ f (x) + f (y) + 2ε.

a∈A

This implies (M-EXC[R]) if α0 = α0 (ε) does not tend to zero as ε → 0 and v = v(ε) remains the same as ε → 0. For the former property, we can take an augmenting path π such that α0 ≥ (x(u) − y(u))/|A|, and for the latter we may take a subsequence of ε that corresponds to a single v. In the discrete case of (2.39), α0 is a positive integer and α = 1 is a valid choice. Hence follows (M-EXC[Z]).

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61

Note 2.20. We prove that g in (2.45) satisfies (SBF[R]) and (TRF[R]) under the assumption that g > −∞. First, (TRF[R]) is obvious from δ(˜ p + α1) = δ p˜. Fix p, q ∈ domR g. For any ε > 0 there exist ηp , ηq , p˜, and q˜ satisfying (2.48) and (2.49). p ∨ q˜) and η∧ = −δ(˜ p ∧ q˜), we have For η∨ = −δ(˜ η∨ (a) = max(˜ p(v), q˜(v)) − max(˜ p(u), q˜(u)) p(v), q˜(v)) − min(˜ p(u), q˜(u)) η∧ (a) = min(˜

(a = (u, v) ∈ A), (a = (u, v) ∈ A).

Hence, for each a ∈ A, there exists λa (0 ≤ λa ≤ 1) such that η∨ (a) = λa ηp (a) + (1 − λa )ηq (a),

η∧ (a) = (1 − λa )ηp (a) + λa ηq (a),

which, together with the convexity of ga , implies that ga (ηp (a)) + ga (ηq (a)) ≥ ga (η∨ (a)) + ga (η∧ (a)). Therefore, we obtain g(p) + g(q) + 2ε ≥



[ga (η∨ (a)) + ga (η∧ (a))] ≥ g(p ∨ q) + g(p ∧ q),

a∈A

which implies (SBF[R]), since ε > 0 is arbitrary. The discrete case (2.43) with ga in (2.40) can be treated similarly.

2.3

Substitutes and Complements in Network Flows

In section 1.3 we explained that submodularity should be compared to convexity. This statement is certainly true for set functions, but, when it comes to functions in real or integer vectors, it is more appropriate to regard convexity and submodularity as mutually independent properties. In this section we address this issue with reference to substitutes and complements in network flows discussed in the literature and show that the concepts of L-convexity and M-convexity help us better understand the relationship between convexity and submodularity.

2.3.1

Convexity and Submodularity

We consider a network flow problem. Let G = (V, A) be a directed graph with vertex set V and arc set A. For each arc a ∈ A, we are given a nonnegative capacity c(a) for flow and a weight w(a) per unit flow. The maximum weight circulation problem is to find a flow ξ = (ξ(a) | a ∈ A) that maximizes the total weight a∈A w(a)ξ(a) subject to the capacity (feasibility) constraint 0 ≤ ξ(a) ≤ c(a)

(a ∈ A)

and the conservation constraint   {ξ(a) | a ∈ δ − v} = 0 {ξ(a) | a ∈ δ + v} −

(2.51)

(v ∈ V ).

(2.52)

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We denote by F the maximum weight of a feasible circulation. Our concern here is how the weight F depends on the problem parameters (w, c). Namely, we are interested in the function F = F (w, c) in w ∈ RA and c ∈ RA + . We first look at convexity and concavity. Proposition 2.21. F is convex in w and concave in c. Proof. F = max{w ξ | N ξ = 0, 0 ≤ ξ ≤ c} is the maximum of linear functions in w and hence convex in w, where N ξ = 0 represents the conservation constraint (2.52). By linear programming duality (see Theorem 3.10 (2)), we obtain an alternative expression F = min{c η | N  p + η ≥ w, η ≥ 0}, which shows the concavity of F in c. We next consider submodularity and supermodularity. A function f : Rn → R ∪ {+∞} is said to be submodular if f (x) + f (y) ≥ f (x ∨ y) + f (x ∧ y)

(x, y ∈ Rn )

(2.53)

(x, y ∈ Rn ),

(2.54)

and supermodular if f (x) + f (y) ≤ f (x ∨ y) + f (x ∧ y)

where x ∨ y and x ∧ y are, respectively, the vectors of componentwise maxima and minima of x and y as defined in (2.16). With the economic terms substitutes and complements we have the following correspondences: f is submodular f is supermodular

⇐⇒ goods are substitutes, ⇐⇒ goods are complements,

where f is interpreted as representing a utility function. Two arcs are said to be parallel if every simple cycle33 containing both of them orients them in the opposite direction, and series if every simple cycle containing both of them orients them in the same direction. A set of arcs is said to be parallel if it consists of pairwise parallel arcs, and series if it consists of pairwise series arcs. With the notation wP = (w(a) | a ∈ P ), cP = (c(a) | a ∈ P ), wS = (w(a) | a ∈ S), and cS = (c(a) | a ∈ S), the following statements hold true, where the proof is given later. Theorem 2.22. Let P be a parallel arc set and S a series arc set. (1) F is submodular in wP and in cP . (2) F is supermodular in wS and in cS . 33 Formally, a simple cycle is an alternating sequence (v , a , v , a , . . . , v 0 1 1 2 k−1 , ak , vk ) of vertices vi (i = 0, 1, . . . , k) and arcs ai (i = 1, . . . , k) such that {∂ + ai , ∂ − ai } = {vi−1 , vi } (i = 1, . . . , k), v0 = vk , and vi = vj (1 ≤ i < j ≤ k).

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Combining Proposition 2.21 and Theorem 2.22 yields that F F F F

is is is is

convex concave convex concave

and and and and

submodular submodular supermodular supermodular

in in in in

wP , cP , wS , cS .

(2.55)

Thus, all combinations of convexity/concavity and submodularity/supermodularity arise in our network flow problem. This demonstrates that convexity and submodularity are mutually independent properties. Although convexity and submodularity are mutually independent, the combinations of convexity/concavity and submodularity/supermodularity in (2.55) are not accidental phenomena but logical consequences that can be explained in terms of L -convexity and M -convexity. The function F is endowed with L -convexity and M -convexity, as follows, where the proof is given in section 2.3.2. Theorem 2.23. Let P be a parallel arc set and S a series arc set. (1) F is L -convex in wP and M -concave in cP . (2) F is M -convex in wS and L -concave in cS . In general, L -convexity implies submodularity by (SBF [R]) in the definition, whereas M -convexity implies supermodularity, as will be shown in Theorem 6.51. Accordingly, L -concavity implies supermodularity and M -concavity submodularity. With the aid of these general results on L -convex and M -convex functions, Theorem 2.23 provides us with a somewhat deeper understanding of (2.55). Namely, it is understood that F F F F

2.3.2

is is is is

L -convex, M -concave, M -convex, L -concave,

hence hence hence hence

convex and submodular, concave and submodular, convex and supermodular, concave and supermodular,

in in in in

wP , cP , wS , cS .

(2.56)

Technical Supplements

This section gives the proof of Theorem 2.23. We start with basic properties of parallel and series arc sets that we use in the proof. Let us call π : A → {0, ±1} a circuit if ∂π = 0 and supp+ (π) ∪ supp− (π) forms a simple cycle. Proposition 2.24. Let π be a circuit. (1) |supp+ (π) ∩ P | ≤ 1 and |supp− (π) ∩ P | ≤ 1 for a parallel arc set P . (2) |supp+ (π) ∩ S| = 0 or |supp− (π) ∩ S| = 0 for a series arc set S. Proposition 2.25. Let S be a series arc set and π1 and π2 be circuits. If supp+ (π1 ) ∩ supp+ (π2 ) ∩ S = ∅, there exists a circuit π such that supp+ (π) ⊆ supp+ (π1 ) ∪ supp+ (π2 ), supp− (π) ⊆ supp− (π1 ) ∪ supp− (π2 ), and supp+ (π) ∩ S = (supp+ (π1 ) ∪ supp+ (π2 )) ∩ S.

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Proof. Suppose a ∈ (supp+ (π2 )\supp+ (π1 ))∩S. By an elementary graph argument we can find a circuit π  such that supp+ (π  ) ⊆ supp+ (π1 ) ∪ supp+ (π2 ), supp− (π  ) ⊆ supp− (π1 ) ∪ supp− (π2 ), and supp+ (π  ) ∩ S ⊇ (supp+ (π1 ) ∩ S) ∪ {a}. Repeating this we can find π. The main technical tool in the proof is the conformal decomposition 34 of a circulation ξ, which is a representation of ξ as a positive sum of circuits conformal to ξ; i.e., m  βi πi , (2.57) ξ= i=1

where βi > 0 and πi : A → {0, ±1} is a circuit with supp+ (πi ) ⊆ supp+ (ξ) and supp− (πi ) ⊆ supp− (ξ) for i = 1, . . . , m. Proof of L -Convexity in wP The L -convexity of F in wP is equivalent to the submodularity of F (w − w0 χP , c) in (wP , w0 ), which in turn is equivalent to F (w + λχa , c) + F (w + μχb , c) ≥ F (w, c) + F (w + λχa + μχb , c),

(2.58)

F (w + λχa , c) + F (w − μχP , c) ≥ F (w, c) + F (w + λχa − μχP , c)

(2.59)

for a, b ∈ P with a = b and λ, μ ∈ R+ . To show (2.58) let ξ and ξ be optimal circulations for w and w + λχa + μχb . We can establish (2.58) by constructing feasible circulations ξa and ξb such that ξa + ξb = ξ + ξ,

λ[ξa (a) − ξ(a)] + μ[ξb (b) − ξ(b)] ≥ 0,

(2.60)

since this implies w + λχa , ξa + w + μχb , ξb ≥ w, ξ + w + λχa + μχb , ξ , where the left-hand side is bounded by F (w + λχa , c) + F (w + μχb , c) and the right-hand side is equal to F (w, c) + F (w + λχa + μχb , c). If ξ(a) ≤ ξ(a), we can take ξa = ξ and ξb = ξ to meet (2.60). If ξ(b) ≤ ξ(b), we can take ξa = ξ and ξb = ξmto meet (2.60). Otherwise, we make use of the conformal decomposition ξ − ξ = i=1 βi πi . Since a ∈ supp+ (ξ − ξ), we may assume πi (a) = 1 for i = 1, . . . ,  and πi (a) = 0 for i =  + 1, . . . , m. We have πi (b) = 0 for i = 1, . . . ,  by Proposition 2.24 (1), since P is parallel and {a, b} ⊆ supp+ (ξ − ξ). Then ξa = ξ + i=1 βi πi and ξb = ξ + m i= +1 βi πi are feasible circulations that satisfy (2.60). To show (2.59) let ξ and ξ be optimal circulations for w and w + λχa − μχP . We can establish (2.59) by constructing feasible circulations ξa and ξP such that ξa + ξP = ξ + ξ,

λ[ξa (a) − ξ(a)] + μ[ξ(P ) − ξP (P )] ≥ 0,

(2.61)

34 More generally, the conformal decomposition is defined for a vector in a subspace in terms of elementary vectors of the subspace; see Iri [94] and Rockafellar [178].

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since this implies w + λχa , ξa + w − μχP , ξP ≥ w, ξ + w + λχa − μχP , ξ . ξP = ξ to meet (2.61). Otherwise, we use If ξ(a) ≤ ξ(a), we can take ξa = ξ and m the conformal decomposition ξ − ξ = i=1 βi πi , in which we assume πi (a) = 1 for i = 1, . . . ,  and πi (a) = 0 for i =  + 1, . . . , m. Since P is parallel, we have |supp− (πi ) ∩ P | ≤ 1 by Proposition 2.24 (1) and hence πi (P ) ≥ 0 for i = 1, . . . , . m Therefore, ξa = ξ + i=1 βi πi and ξP = ξ + i= +1 βi πi are feasible circulations with the properties in (2.61). Proof of M -Concavity in cP We prove the M -concavity of F in cP by establishing (M -EXC[R]) for −F as a function in cP . In our notation this reads as follows:    Let c1 , c2 ∈ RA + be capacities with c1 (a ) = c2 (a ) for all a ∈ A \ P . For each a ∈ supp+ (c1 − c2 ), there exist b ∈ supp− (c1 − c2 ) ∪ {0} and a positive number α0 such that

F (w, c1 ) + F (w, c2 ) ≤ F (w, c1 − α(χa − χb )) + F (w, c2 + α(χa − χb )) for all α ∈ [0, α0 ]R , where χ0 = 0. Let ξ1 and ξ2 be optimal circulations for c1 and c2 , respectively. We shall find α0 > 0 and b ∈ supp− (c1 − c2 ) ∪ {0} such that, for any α ∈ [0, α0 ]R , there exist circulations ξ1 and ξ2 such that ξ1 + ξ2 = ξ1 + ξ2 ,

0 ≤ ξ1 ≤ c1 − α(χa − χb ),

0 ≤ ξ2 ≤ c2 + α(χa − χb ). (2.62)

If ξ1 (a) < c1 (a), we can take α0 = c1 (a) − ξ1 (a), b = 0, ξ1 = ξ1 , and ξ2 = ξ2 to meet (2.62). Suppose ξ1 (a) = c1 (a). We have ξ1 (a) = c1 (a) > c2 (a) ≥ ξ2 (a). Let π be a circuit such that a ∈ supp+ (π) ⊆ supp+ (ξ1 − ξ2 ) and supp− (π) ⊆ supp− (ξ1 − ξ2 ). Since P is parallel and a ∈ supp+ (π), we have supp+ (π) ∩ P = {a} and |supp− (π) ∩ P | ≤ 1 by Proposition 2.24 (1). If |supp− (π) ∩ P | = 1, define b by {b} = supp− (π) ∩ P ; otherwise put b = 0. We can take α0 > 0 such that α0 ≤ |ξ1 (a ) − ξ2 (a )| for all a ∈ supp+ (π) ∪ supp− (π). Then ξ1 = ξ1 − απ and ξ2 = ξ2 + απ satisfy (2.62) if 0 ≤ α ≤ α0 . Proof of M -Convexity in wS We prove the M -convexity of F in wS by establishing (M -EXC[R]). In our notation this reads as follows: Let w1 , w2 ∈ RA be weight vectors with w1 (a ) = w2 (a ) for all a ∈ A\S. For each a ∈ supp+ (w1 −w2 ), there exist b ∈ supp− (w1 −w2 )∪{0} and a positive number α0 such that F (w1 , c) + F (w2 , c) ≥ F (w1 − α(χa − χb ), c) + F (w2 + α(χa − χb ), c) for all α ∈ [0, α0 ]R , where χ0 = 0.

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Let ξ1 and ξ2 be optimal circulations for w1 and w2 , respectively, with ξ1 (a) minimum and ξ2 (a) maximum. Proposition 2.26. There exists α0 > 0 such that ξ1 is optimal for w1 − αχa and ξ2 is optimal for w2 + αχa for all α ∈ [0, α0 ]R . Proof. For any circuit π such that π(a) = −1 and 0 ≤ ξ1 + βπ ≤ c for some β > 0, we have w1 , ξ1 + βπ < w1 , ξ1 by the choice of ξ1 . Let α1 > 0 be the minimum of − w1 , π over all such circuits π. Then ξ1 is optimal for w1 − αχa for all α ∈ [0, α1 ]R , since w1 − αχa , ξ1 + βπ ≤ w1 − αχa , ξ1 for any β > 0 and circuit π such that 0 ≤ ξ1 + βπ ≤ c. Similarly, let α2 > 0 be the minimum of − w2 , π

over all circuits π such that π(a) = 1 and 0 ≤ ξ2 + βπ ≤ c for some β > 0. Then ξ2 is optimal for w2 + αχa for all α ∈ [0, α2 ]R . Put α0 = min(α1 , α2 ). If ξ1 (a) ≥ ξ2 (a), we can take b = 0 in (M -EXC[R]), since F (w1 , c) + F (w2 , c) = w1 , ξ1 + w2 , ξ2

≥ w1 − αχa , ξ1 + w2 + αχa , ξ2 = F (w1 − αχa , c) + F (w2 + αχa , c), where the last equality is by Proposition 2.26. In what follows we assume ξ1 (a) < ξ2 (a). By Proposition 2.24 (2), we can impose further conditions on ξ1 and ξ2 that, for each b ∈ S \ {a}, ξ1 (b) is maximum among all optimal ξ1 for w1 with ξ1 (a) minimum, and ξ2 (b) is minimum among all optimal ξ2 for w2 with ξ2 (a) maximum. Proposition 2.27. There exists α0 > 0 such that ξ1 is optimal for w1 − α(χa − χb ) and ξ2 is optimal for w2 + α(χa − χb ) for all b ∈ S \ {a} and for all α ∈ [0, α0 ]R . Proof. For any circuit π such that π(a) − π(b) = −1 for some b ∈ S \ {a} and 0 ≤ ξ1 + βπ ≤ c for some β > 0, we have w1 , ξ1 + βπ < w1 , ξ1 by the choice of ξ1 . Let α1 > 0 be the minimum of − w1 , π over all such circuits π. Then ξ1 is optimal for w1 − α(χa − χb ) for all α ∈ [0, α1 ]R . Similarly, let α2 > 0 be the minimum of − w2 , π over all circuits π such that π(a) − π(b) = 1 for some b ∈ S \ {a} and 0 ≤ ξ2 + βπ ≤ c for some β > 0. Then ξ2 is optimal for w2 + α(χa − χb ) for all α ∈ [0, α2 ]R . Put α0 = min(α1 , α2 ). Proposition 2.27 implies F (w1 , c) + F (w2 , c) − F (w1 − α(χa − χb ), c) − F (w2 + α(χa − χb ), c) = w1 , ξ1 + w2 , ξ2 − w1 − α(χa − χb ), ξ1 − w2 + α(χa − χb ), ξ2

(2.63) = α[(ξ2 (b) − ξ1 (b)) − (ξ2 (a) − ξ1 (a))]. We want to find b ∈ supp− (w1 − w2 ) for which (2.63) is nonnegative. We make use of the conformal decomposition ξ2 − ξ1 = m i=1 βi πi . Since S is series, we may assume, by Proposition 2.25, that a ∈ supp+ (π1 ) ∩ S ⊆ supp+ (π2 ) ∩ S ⊆ · · · ⊆ supp+ (π ) ∩ S

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and πi (a) = 0 for i =  + 1, . . . , m. Then supp− (πi ) ∩ S = ∅ for i = 1, . . . , . Proposition 2.28. There exists b ∈ (supp+ (π1 ) ∩ S) ∩ supp− (w1 − w2 ). Proof. We have w1 , π1 ≤ 0, since ξ1 is optimal for w1 and 0 ≤ ξ1 + β1 π1 ≤ c. Similarly, we have w2 , −π1 ≤ 0. Hence,   (w1 (b) − w2 (b))π1 (b) = (w1 (b) − w2 (b)). 0 ≥ w1 − w2 , π1 = b∈supp+ (π1 )∩S

b∈S

Since w1 (a) − w2 (a) > 0 in this summation, we must have w1 (b) − w2 (b) < 0 for some b ∈ supp+ (π1 ) ∩ S. For b ∈ (supp+ (π1 ) ∩ S) ∩ supp− (w1 − w2 ) in Proposition 2.28, we have ξ2 (b) − ξ1 (b) =



βi +

i=1

m 

βi πi (b) ≥

i= +1



βi = ξ2 (a) − ξ1 (a),

i=1

which shows the nonnegativity of (2.63). Proof of L -Concavity in cS The L -concavity of F in cS is equivalent to the supermodularity of F (w, c − c0 χS ) in (cS , c0 ), which in turn is equivalent to F (w, c + λχa ) + F (w, c + μχb ) ≤ F (w, c) + F (w, c + λχa + μχb ), F (w, c + λχa ) + F (w, c − μχS ) ≤ F (w, c) + F (w, c + λχa − μχS )

(2.64) (2.65)

for a, b ∈ S with a = b and λ, μ ∈ R+ . To show (2.64), let ξa and ξb be optimal circulations for c + λχa and c + μχb . We can establish (2.64) by constructing circulations ξ and ξ such that ξ + ξ = ξa + ξb ,

0 ≤ ξ ≤ c,

0 ≤ ξ ≤ c + λχa + μχb .

(2.66)

If ξa (a) ≤ c(a), we can take ξ = ξa and ξ = ξb to meet (2.66). If ξb (b) ≤ c(b), we can take ξ = ξb and ξ = ξa to meet (2.66). Otherwise, we have ξa (a) > c(a) ≥ ξb (a) ξb ) and b ∈ supp− (ξa − ξb ). and ξa (b) ≤ c(b) < ξb (b), and therefore a ∈ supp+ (ξa − m We make use of the conformal decomposition ξa − ξb = i=1 βi πi , where we assume πi (a) = 1 for i = 1, . . . ,  and πi (a) = 0 for i =  + 1, . . . , m. We have πi (b) = 0 for i = 1, . . . ,  by Proposition 2.24 (2), since S is series and a ∈ supp+ (ξa − ξb ) and b ∈ supp− (ξa − ξb ). Then ξ = ξa − i=1 βi πi and ξ = ξb + i=1 βi πi satisfy (2.66). To show (2.65), let ξa and ξS be optimal circulations for c + λχa and c − μχS . We can establish (2.65) by constructing circulations ξ and ξ such that ξ + ξ = ξa + ξS ,

0 ≤ ξ ≤ c,

0 ≤ ξ ≤ c + λχa − μχS .

(2.67)

If ξa (a) ≤ c(a), we can take ξ = ξa and ξ = ξS to meet (2.67). Otherwise, we have ξa (a) > c(a) ≥ ξS (a), and therefore a ∈ supp+ (ξa − ξS ). We use the conformal

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decomposition ξa −ξS = 2.25 that

m i=1

βi πi . Since S is series, we may assume by Proposition

a ∈ supp+ (π1 ) ∩ S ⊆ supp+ (π2 ) ∩ S ⊆ · · · ⊆ supp+ (π ) ∩ S and πi (a) = 0 for i =  + 1, . . . , m. Then supp− (πi ) ∩ S = ∅ for i = 1, . . . , . Noting k k be the smallest integer with i=1 βi ≥ i=1 βi = ξa (a) − ξS (a) ≥ ξa (a) − c(a), let k−1 k−1 ξa (a)−c(a) and define β  = [ξa (a)−c(a)]− i=1 βi . Then ξ = ξa − i=1 βi πi −β  πk k−1 k−1 and ξ = ξS + i=1 βi πi + β  πk satisfy (2.67), since ξ(a) = ξa (a) − i=1 βi − β  = k−1 c(a), ξ(a) = ξS (a) + i=1 βi + β  = ξS (a) + ξa (a) − c(a) ≤ c(a) + λ − μ, and, for any b ∈ supp+ (πk ) ∩ S, we have ξ(b) = ξS (b) + +

k−1 

βi πi (b) + β = ξS (b) +

i=1

= ξS (b) +





k−1  i=1

,

βi πi (b) +

+ 

, βi + ξS (a) − c(a)

i=k

βi πi (b) + ξS (a) − c(a) ≤ ξa (b) + ξS (a) − c(a) ≤ c(b) − μ.

i=1

This completes the proof of Theorem 2.23.

2.4

Matroids

In section 1.3.2 we introduced the concept of base polyhedra in terms of an abstract exchange axiom and mentioned that a matroid can be identified with a base polyhedron having vertices of {0, 1}-vectors. To compensate for such an abstract definition of matroids, we explain here some linear-algebraic facts behind the abstract axioms. The key is the Grassmann–Pl¨ ucker relation for determinants, qualitative analyses of which lead to the concepts of matroids and valuated matroids.

2.4.1

From Matrices to Matroids

Suppose we are given a matrix, say, 1 1 A= 0 0

2 0 1 0

3 0 0 1

4 1 1 0

5 0 = [a1 , . . . , a5 ] , 1 1

(2.68)

where a1 , . . . , a5 ∈ R3 denote the column vectors. Let V denote the set of its column indices; we have V = {1, . . . , 5} in our example. The concept of matroids is derived from a combinatorial consideration of linear dependence among column vectors. We say that a subset J of V is independent if the corresponding column vectors {aj | j ∈ J} are linearly independent. Since a subset of an independent set is obviously independent, we may focus on maximal independent sets (maximal with respect to set inclusion). A maximal independent

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69

set is called a base and the family of bases (or base family) is denoted by B. In our example we have B = {{1, 2, 3}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {1, 4, 5}, {2, 3, 4}, {2, 4, 5}, {3, 4, 5}}. The base family B has a remarkable combinatorial property: (B) For any J, J  ∈ B and any i ∈ J \ J  , there exists j ∈ J  \ J such that J − i + j ∈ B and J  + i − j ∈ B, / J  } and J − i + j and J  + i − j are shorthand where J \ J  = {k | k ∈ J, k ∈ notations for (J \ {i}) ∪ {j} and (J  ∪ {i}) \ {j}, respectively. For instance, take J = {1, 2, 3} and J  = {3, 4, 5} in our example. For i = 1 we can take j = 4 to obtain J − i + j = {4, 2, 3} ∈ B and J  + i − j = {3, 1, 5} ∈ B. The choice of j = 5 is not valid, since J − i + j = {5, 2, 3} ∈ / B. The property (B) above is called the (simultaneous) exchange property. For the proof of the exchange property (B) we need to introduce the Grassmann– Pl¨ ucker relation, a fundamental fact in linear algebra. For J ⊆ V we denote35 the determinant of the submatrix A[J] = (aj | j ∈ J) by det A[J]. The Grassmann– Pl¨ ucker relation is an identity  det A[J − i + j]·det A[J  + i − j] (2.69) det A[J]·det A[J  ] = j∈J  \J

valid for any J, J  ⊆ V and any i ∈ J \ J  (the proof is sketched in Note 2.30). The notation det A[J − i + j] here means the determinant of A[J] with column i replaced with column j. To prove (B), suppose that J, J  ∈ B. Then the left-hand side of (2.69) is distinct from zero, and therefore, there exists a nonzero term, say, indexed by j ∈ J  \ J, in the summation on the right-hand side of (2.69). Then we have det A[J − i + j] = 0 and det A[J  + i − j] = 0; i.e., J − i + j ∈ B and J  + i − j ∈ B. This proves (B). We emphasize that the exchange property (B) is derived from the Grassmann–Pl¨ ucker relation by a qualitative consideration that distinguishes between zero and nonzero, while disregarding the numerical information. An alternative representation of linear independence among column vectors is given by the rank function ρ : 2V → Z defined by ρ(X) = rank {aj | j ∈ X}

(X ⊆ V ).

(2.70)

The rank function has the following properties (proved in Note 2.31): (R1) 0 ≤ ρ(X) ≤ |X|, (R2) X ⊆ Y =⇒ ρ(X) ≤ ρ(Y ), (R3) ρ(X) + ρ(Y ) ≥ ρ(X ∪ Y ) + ρ(X ∩ Y ). 35 We consider J such that A[J] is square and implicitly assume an ordering of the elements of J, which affects the sign of the determinant.

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The third property (R3) shows that ρ is a submodular function. The rank function and the base family determine each other by ρ(X) = max{|X ∩ J| | J ∈ B} (X ⊆ V ),

(2.71)

B = {J ⊆ V | ρ(J) = |J| = ρ(V )}.

(2.72)

From a given matrix we have thus derived a set family B ⊆ 2V with property (B) and a set function ρ : 2V → Z with property (R) (i.e., (R1) to (R3)), where V is the set of columns of the given matrix. It is all-important to realize that the properties (B) and (R) are stated without reference to the given matrix and, as such, they make sense as properties of a set family and a set function in general. Matroid theory adopts these properties as abstract axioms and studies the combinatorial structure implied by these axioms (and nothing else). It turns out that a set family B satisfying axiom (B) and a set function ρ satisfying axiom (R) are equivalent to each other. More precisely, we have the following statement. Theorem 2.29. The class of set functions ρ : 2V → Z satisfying (R1), (R2), and (R3) and the class of nonempty families B ⊆ 2V satisfying (B) are in one-to-one correspondence through the mutually inverse mappings (2.71) and (2.72). In this sense, the two objects B and ρ represent one and the same combinatorial structure, which is called a matroid . That is, a matroid is a pair (V, B) of a finite set V and a family B of subsets of V that satisfies (B), or, alternatively, a matroid is a pair (V, ρ) of a finite set V and a set function ρ on V that satisfies (R). We may also denote a matroid by a triple (V, B, ρ). The set V is called the ground set , B is the base family, and ρ is the rank function of the matroid. A member of B is a base and a subset of a base is an independent set . Though defined by such simple axioms, the concept of matroids is fundamental and fruitful in studying combinatorial structures. The exchange property (B) above is the germ of (B-EXC[Z]) treated in section 1.3.2 and (R3) is the submodularity featured in section 1.3.1. The one-to-one correspondence between B and ρ stated in Theorem 2.29 above is the germ of Theorem 1.9 that establishes the equivalence between exchangeability and submodularity. Note 2.30. The idea of the proof of the Grassmann–Pl¨ ucker relation (2.69) is indicated here for a 3 × 5 matrix A=

-

a1

a2

a3

a4

a5

.

(a1 , . . . , a5 ∈ R3 ) and for J = {1, 2, 3}, J  = {3, 4, 5}, and i = 1. Consider a 6 × 6 matrix   a1 a2 a3 a3 a4 a5 A˜ = , a1 0 0 a3 a4 a5

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where aj = aj (j = 1, 3, 4, 5). The generalized Laplace expansion applied to det A˜ yields det A˜ = det[a1 , a2 , a3 ] · det[a3 , a4 , a5 ] − det[a4 , a2 , a3 ] · det[a3 , a1 , a5 ] − det[a5 , a2 , a3 ] · det[a3 , a4 , a1 ]. On the other hand, subtracting the lower half (three rows) from the upper half (three rows) of A˜ yields   0 a2 a3 0 0 0 , a1 0 0 a3 a4 a5 which is obviously singular. Hence, det A˜ = 0, and det[a1 , a2 , a3 ] · det[a3 , a4 , a5 ] = det[a4 , a2 , a3 ] · det[a3 , a1 , a5 ] + det[a5 , a2 , a3 ] · det[a3 , a4 , a1 ], establishing (2.69) for J = {1, 2, 3}, J  = {3, 4, 5}, and i = 1. Note 2.31. We prove (R1), (R2), and (R3) for the rank function (2.70) associated with a matrix. (R1) and (R2) are obvious. To prove (R3), let {aj | j ∈ JXY } (where JXY ⊆ X ∩Y ) be a base of {aj | j ∈ X ∩Y }. There exists some JX ⊆ X \Y such that {aj | j ∈ JXY ∪ JX } is a base of {aj | j ∈ X}, since any set of independent vectors can be augmented to a base. For the same reason, there exists some JY ⊆ Y \ X such that {aj | j ∈ JXY ∪ JX ∪ JY } is a base of {aj | j ∈ X ∪ Y }. Then we have |JXY | = ρ(X ∩ Y ), |JXY | + |JX | = ρ(X), |JXY | + |JX | + |JY | = ρ(X ∪ Y ), and |JXY | + |JY | ≤ ρ(Y ), where the last inequality is due to the independence of the vectors indexed by JXY ∪ JY . Hence follows (R3).

2.4.2

From Polynomial Matrices to Valuated Matroids

In section 2.4.1 we abstracted the axiom of a matroid from the Grassmann–Pl¨ ucker relation for matrices. A similar argument for polynomial matrices leads us to the concept of valuated matroids, which may be thought of as discrete concave functions. Suppose we are given a polynomial matrix in variable s, say,

A(s) =

1 s+1 1

2 s 1

3 1 1

4 0 , 1

(2.73)

with the column set V = {1, 2, 3, 4}. Since the determinant det A[J] is a polynomial in s, we can talk of its degree, which we denote by ω(J): ω(J) = deg det A[J]

(J ⊆ V ),

(2.74)

where we put ω(J) = −∞ if det A[J] = 0 or A[J] is nonsquare (when det A[J] is not defined). Using the notation B for the family of bases we have ω(J) = −∞ ⇐⇒ J ∈ B.

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We now look at the Grassmann–Pl¨ ucker relation (2.69) with respect to the degree in s. For J, J  ∈ B, the degree of the left-hand side of (2.69) is equal to ω(J) + ω(J  ). This implies that at least one term on the right-hand side must have degree not lower than this. Hence, the function ω has the following property: (VM) For any J, J  ∈ B and any i ∈ J \ J  , there exists j ∈ J  \ J such that J − i + j ∈ B, J  + i − j ∈ B, and ω(J) + ω(J  ) ≤ ω(J − i + j) + ω(J  + i − j). The inequality can be strict due to possible cancellations of the highest degree terms on the right-hand side of (2.69). In our example (2.73) we have det A[{1, 2}] = det A[{3, 4}] = 1, and hence ω(J) = ω(J  ) = 0 for J = {1, 2} and J  = {3, 4}. For i = 1 ∈ J \ J  we can choose j = 3 ∈ J  \ J, for which ω(J − i + j) + ω(J  + i − j) = 2. The concept of valuated matroids is obtained by adopting (VM) as an axiom. Namely, a valuated matroid is a pair (V, ω) of a finite set V and a set function ω : 2V → R ∪ {−∞} that satisfies (VM), where it is assumed that B = {J ⊆ V | ω(J) = −∞}

(2.75)

is nonempty. Not surprisingly, valuated matroids are closely related to matroids. First, (VM) for ω implies (B) for B. This means that (V, B) is a matroid if (V, ω) is a valuated matroid. Accordingly, B is called the base family of the valuated matroid (V, ω). It is also said that ω is a valuation of the matroid (V, B). Next, the maximizers of ω form the base family of a matroid. This is because, for two maximizers J and J  with ω(J) = ω(J  ) = max ω, we have ω(J − i + j) = ω(J  + i − j) = max ω in (VM), which means (B) for the family of maximizers of ω. Furthermore, for any p = (p(i) | i ∈ V ) ∈ RV , the function ω[−p] : 2V → R∪{−∞} defined by  ω[−p](J) = ω(J) − p(j) (2.76) j∈J

is a valuated matroid, and therefore, the maximizers of ω[−p] form the base family of a matroid for each p ∈ RV . This property, in turn, characterizes a valuated matroid as follows. Theorem 2.32. Let (V, B) be a matroid with ground set V and base family B. A function ω : B → R is a valuation if and only if for any p : V → R the maximizers of ω[−p] form the base family of a matroid. Proof. This is a special case of Theorem 6.30 to be shown later. This theorem seems to suggest that we should compare valuated matroids to concave functions and matroids to convex sets. Here is a connection of valuated

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matroids to M-convexity. A set function ω : 2V → R ∪ {−∞} can be identified with a function f : ZV → R ∪ {+∞} with domZ f ⊆ {0, 1}V by  −ω(J) (x = χJ , J ∈ B), (2.77) f (x) = +∞ (otherwise), where domZ f = {χJ | J ∈ B} with B in (2.75). It is easy to observe that (VM) for ω is equivalent to (M-EXC[Z]) for f . That is, ω is a valuated matroid if and only if f is an M-convex function. For instance, the valuated matroid (V, ω) associated with the polynomial matrix in (2.73) can be identified with an M-convex function ⎧ (x ∈ B0 ), ⎨ 0 −1 (x ∈ B \ B0 ), f (x) = ⎩ +∞ (x ∈ ZV \ B), where B = {(1, 1, 0, 0), (0, 0, 1, 1), (1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 1, 0), (0, 1, 0, 1)} and B0 = {(1, 1, 0, 0), (0, 0, 1, 1)}. In parallel with the generalization of the base family to a valuation ω, the rank function of a matroid can be generalized as follows. Assuming that ω is an integer-valued valuation, we define a function g : ZV → Z ∪ {+∞} by ⎫ ) ⎧ ) ⎬ ⎨  ) (p ∈ ZV ). p(j))) J ∈ B (2.78) g(p) = max ω(J) + ⎭ ⎩ ) j∈J

In our example of (2.73) we have g(p) = max(p(1) + p(2), p(3) + p(4), p(1) + p(3) + 1, p(1) + p(4) + 1, p(2) + p(3) + 1, p(2) + p(4) + 1). As is shown in Note 2.34, the function g is submodular over the integer lattice: g(p) + g(q) ≥ g(p ∨ q) + g(p ∧ q)

(p, q ∈ ZV ),

(2.79)

which is a generalization of the submodularity (R3) of the rank function of a matroid. The connection to a matroid rank function is conspicuous in the special case where ω(J) = 0 for all J ∈ B. Then we have g(χX ) = max{|X ∩ J| | J ∈ B} = ρ(X)

(X ⊆ V )

by (2.78) and (2.71), where ρ is the rank function of the underlying matroid (V, B). Note 2.33. We have started with a polynomial matrix to define a function ω with property (VM). As is evident from the proof, the same construction works for a matrix over a non-Archimedian valuated field (van der Waerden [205]). The name “valuated matroid” comes from this fact.

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Note 2.34. The submodularity (2.79) of the function g in (2.78) can be proved as follows. First we show g(χi ) + g(χj ) ≥ g(0) + g(χi + χj )

(i = j),

(2.80)

which is a special case of (2.79) for p = χi and q = χj . Take I, J ∈ B with g(0) = ω(I) and g(χi + χj ) = ω(J) + |J ∩ {i, j}|. If |J ∩ {i, j}| ≤ 1, we have g(χi + χj ) = max(g(χi ), g(χj )), which implies (2.80). The case of |J ∩ {i, j}| = |I ∩ {i, j}| = 2 is also easy. Suppose that |J ∩ {i, j}| = 2 and |I ∩ {i, j}| ≤ 1, and assume j ∈ J \I without loss of generality. By (VM), there exists k ∈ I \J for which ω(I) + ω(J) ≤ ω(I + j − k) + ω(J − j + k). This establishes (2.80), since ω(I) = g(0), ω(J) = g(χi + χj ) − 2, ω(I + j − k) ≤ g(χj ) − 1, and ω(J − j + k) ≤ g(χi ) − 1. For p = (p ∧ q) + χi and q = (p ∧ q) + χj (i = j), the same argument applies to ω  (J) = ω(J) + j∈J (p ∧ q)(j) to prove (2.79). The general case can be treated by induction on ||p − q||1 ; we may assume supp+ (p − q) = ∅ and add the inequalities (2.79) for (p − χi , q) and for (p, (p ∨ q) − χi ) with i ∈ supp+ (p − q).

Bibliographical Notes M-matrices are well-studied objects in applied mathematics and a comprehensive treatment of their mathematical properties can be found in Berman–Plemmons [9]. The connection of symmetric M-matrices to L-/M-convex quadratic functions presented in section 2.1 is mostly based on Murota [147], whereas the general case described in section 2.1.4 is due to Murota–Shioura [155]. See Fukushima–Oshima– Takeda [71] and Doyle–Snell [39] for connections to probability theory. For network flow problems in combinatorial optimization, Ford–Fulkerson [53] is the classic, whereas Ahuja–Magnanti–Orlin [1] describes recent algorithmic developments. Thorough treatments of the network flow problem on the basis of convex analysis can be found in Iri [94] and Rockafellar [178], the former putting more emphasis on physical issues and the latter more mathematical. In particular, the functions f and g in (2.37) and (2.38) are considered in the case of |T | = 2 in [94] and [178]. The variational formulations are also discussed in Brayton–Moser [19] and Clay [25]. The terminologies of current potential and voltage potential are taken from [19], though they seem to be more often called content and cocontent. M-convexity and L-convexity in the network flow problem are pointed out in Murota [140], [141], [142]. Substitutes and complements in network flows are discussed in Gale–Politof [73], Granot–Veinott [79], and Shapley [184], [185]. In particular, Theorem 2.22 is due to Gale–Politof [73]. The connection to M-convexity and L-convexity (Theorem 2.23) is due to Murota–Shioura [158]. A number of books on matroids are available: Oxley [170], Welsh [211], and White [212], [213], [214] are standard mathematical textbooks; Recski [175] realizes a successful balance between theory and application; and Murota [146] emphasizes linear-algebraic motivations. For optimization on matroids, see, e.g., Cook– Cunningham–Pulleyblank–Schrijver [26], Faigle [48], Korte–Vygen [115], Lawler [119], and Schrijver [183]. Key papers in the development of matroid theory, including Whitney [218], are collected in Kung [116]. Nakasawa [164] gives simple

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exchange axioms for matroids. The simultaneous exchange property for matroids is due to Brualdi [20]. The concept of valuated matroids is due to Dress–Wenzel [41], [42]. Chapter 5 of [146] is a systematic presentation of the theory of valuated matroids including duality. Circuit axioms are investigated in Murota–Tamura [159] and constrained optimizations in Alth¨ofer–Wenzel [2]. Oriented matroids are another ramification of matroids, for which a comprehensive monograph of Bj¨ orner– Las Vergnas–Sturmfels–White–Ziegler [16] is available.

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Chapter 3

Convex Analysis, Linear Programming, and Integrality

This chapter provides technical elements that are needed in subsequent chapters. Some basic facts in convex analysis and linear programming are given in the first two sections. The following two sections address integrality issues, i.e., integrality for a pair of integral polyhedra and the concept of integrally convex functions.

3.1

Convex Analysis

A minimal set of prerequisites from convex analysis is given in this section, while the reader is referred to the textbooks listed in the bibliographical notes at the end of this chapter for comprehensive accounts. For two vectors a = (a(i))ni=1 and b = (b(i))ni=1 ∈ (R ∪ {±∞})n we define [a, b] = [a, b]R = {x ∈ Rn | a(i) ≤ x(i) ≤ b(i) (i = 1, . . . , n)}, (a, b) = (a, b)R = {x ∈ Rn | a(i) < x(i) < b(i) (i = 1, . . . , n)},

(3.1) (3.2)

where, if a(i) = −∞, for example, a(i) ≤ x(i) is to be understood as −∞ < x(i). Sets such as [a, b] and (a, b) are referred to as closed intervals and open intervals, respectively. For a function f : Rn → R ∪ {±∞} we define dom f = domR f = {x ∈ Rn | −∞ < f (x) < +∞},

(3.3)

which is the effective domain of f . A function f : Rn → R ∪ {+∞} is said to be convex if it satisfies λf (x) + (1 − λ)f (y) ≥ f (λx + (1 − λ)y)

(x, y ∈ Rn , 0 ≤ λ ≤ 1).

(3.4)

Note that −∞ is excluded from the possible function values of a convex function and that the inequality (3.4) is satisfied, by convention, if it is in the form of +∞ ≤ +∞. A convex function with a nonempty effective domain is called a proper convex function. A function is strictly convex if it satisfies (3.4) with strict 77

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inequalities, i.e., if λf (x) + (1 − λ)f (y) > f (λx + (1 − λ)y)

(x, y ∈ dom f, 0 < λ < 1).

(3.5)

A function h : Rn → R ∪ {−∞} is concave if −h is convex, i.e., if λh(x) + (1 − λ)h(y) ≤ h(λx + (1 − λ)y)

(x, y ∈ Rn , 0 ≤ λ ≤ 1).

(3.6)

A set S ⊆ Rn is called convex if it satisfies the condition x, y ∈ S, 0 ≤ λ ≤ 1 =⇒ λx + (1 − λ)y ∈ S,

(3.7)

where an empty set is a convex set. A set S is a cone if it satisfies x ∈ S, λ > 0 =⇒ λx ∈ S.

(3.8)

A convex cone is a cone that is convex and a set S is a convex cone if and only if it satisfies the condition x, y ∈ S, λ, μ > 0 =⇒ λx + μy ∈ S.

(3.9)

A convex polyhedron is a typical convex set S described by a finite number of linear inequalities as ⎧ ) ⎫ ) n ⎨ ⎬  ) S = x ∈ Rn )) aij x(j) ≤ bi (i = 1, . . . , m) , (3.10) ⎩ ⎭ ) j=1 where aij ∈ R and bi ∈ R (i = 1, . . . , m, j = 1, . . . , n). If bi = 0 for all i, then S is a convex cone. For a finite number of points x1 , . . . , xm in a set S, a point represented as λ1 x1 + · · · + λm xm

(3.11)

with nonnegative coefficients λi (1 ≤ i ≤ m) with unit sum ( m i=1 λi = 1) is called a convex combination of those points. The convex closure of S, denoted as S, is defined to be the set of all possible convex combinations of a finite number of points of S. If S is convex, any convex combination of any finite set of points of S belongs to S, and vice versa, and therefore S is convex if and only if S = S. For a set S, the intersection of all the convex sets containing S is the smallest convex set containing S, which is called the convex hull of S. The convex hull of S coincides with the convex closure of S. The convex hull of a set S is not necessarily closed (in the topological sense). The smallest closed convex set containing S is called the closed convex hull of S. For a finite set S, the convex hull is always closed. The affine hull of a set S is defined to be the smallest affine set (a translation of a linear space) containing S and is denoted by aff S. The relative interior of S, denoted as ri S, is the set of points x ∈ S such that {y ∈ Rn | ||y − x|| < ε} ∩ aff S

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is contained in S for some ε > 0. In other words, the relative interior of S is the set of the interior points of S with respect to the topology induced from aff S. We have so far defined convex functions and convex sets independently, but they are actually closely related to each other. The indicator function of a set S ⊆ Rn is a function δS : Rn → {0, +∞} defined by  0 (x ∈ S), δS (x) = (3.12) +∞ (x ∈ / S). Then, as is easily seen, S is a convex set ⇐⇒ δS is a convex function.

(3.13)

This shows how the concept of convex sets can be defined in terms of that of convex functions. Conversely, convex functions can be defined in terms of convex sets. The epigraph of a function f : Rn → R ∪ {+∞}, denoted as epi f , is the set of points in Rn × R lying above the graph of Y = f (x). Namely, epi f = {(x, Y ) ∈ Rn+1 | Y ≥ f (x)}.

(3.14)

f is a convex function ⇐⇒ epi f is a convex set,

(3.15)

Then we have

which shows that the convexity concept for functions can be induced from that for sets. In passing, we mention that a function f is said to be closed convex if epi f is a closed convex set in Rn+1 . A (global ) minimizer of f is a point x such that f (x) ≤ f (y) for all y. The set of the minimizers of f , denoted by arg min f = {x ∈ Rn | f (x) ≤ f (y) (∀ y ∈ Rn )},

(3.16)

is a convex set for a convex function f . A global minimizer of a convex function can be characterized by local minimality (Theorem 1.1). For a family of convex functions {fk | k ∈ K}, indexed by K, the pointwise maximum, f (x) = sup{fk (x) | k ∈ K}, is again a convex function, where the index set K here may possibly be infinite. In particular, the maximum of a finite or infinite number of affine functions f (x) = sup{αk + pk , x | k ∈ K}

(3.17)

is a convex function, where αk ∈ R and pk ∈ Rn for k ∈ K and p, x =

n 

p(i)x(i)

(3.18)

i=1

designates the inner product 36 of p = (p(i))ni=1 and x = (x(i))ni=1 . 36 More precisely, p, x is not so much an inner product as a pairing, since p and x belong to different (mutually dual) spaces.

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A function defined on Rn is said to be polyhedral convex if its epigraph is a convex polyhedron in Rn+1 . A polyhedral convex function is exactly such a function that can be represented as the maximum of a finite number of affine functions (i.e., (3.17) with finite K) on an effective domain represented as (3.10). In the case of n = 1 (univariate case), a polyhedral convex function is nothing but a convex piecewise linear function consisting of a finite number of linear pieces. We denote by C[R → R] the family of univariate polyhedral convex functions. The sum of two functions fk : Rn → R ∪ {+∞} (k = 1, 2) is a function f1 + f2 : Rn → R ∪ {+∞} defined naturally by (f1 + f2 )(x) = f1 (x) + f2 (x)

(x ∈ Rn )

(3.19)

and their infimal convolution is a function f1 2 f2 : Rn → R ∪ {±∞} defined by (f1 2 f2 )(x) = inf{f1 (x1 ) + f2 (x2 ) | x = x1 + x2 , x1 , x2 ∈ Rn }

(x ∈ Rn ). (3.20)

The sum of two convex functions is convex, and the infimal convolution of two convex functions is convex if it does not take the value of −∞. If f1 and f2 are the indicator functions of sets S1 and S2 , then f1 + f2 and f1 2 f2 are the indicator functions of the intersection S1 ∩ S2 and the Minkowski sum S1 + S2 , respectively, where (3.21) S1 + S2 = {x1 + x2 | x1 ∈ S1 , x2 ∈ S2 }. Modifying a function by a linear function is a fundamental operation. For a function f and a vector p, we denote by f [−p] the function defined by f [−p](x) = f (x) − p, x

(x ∈ Rn ).

(3.22)

This is a convex function for f convex. The subdifferential of a function f at a point x ∈ dom f is defined to be the set (3.23) ∂R f (x) = {p ∈ Rn | f (y) − f (x) ≥ p, y − x (∀ y ∈ Rn )}. Note that p ∈ ∂R f (x) if and only if x ∈ arg min f [−p]. Being the intersection of (infinitely many) half-spaces indexed by y, ∂R f (x) is convex (possibly empty) for any f and any x. The set ∂R f (x) is nonempty for f convex and x in the relative interior of dom f . An element of ∂R f (x) is called a subgradient of f at x. If f is convex and differentiable at x, the subdifferential ∂R f (x) consists of a single element, which is the gradient ∇f = (∂f /∂x(i))ni=1 of f at x. The directional derivative of a function f at a point x ∈ dom f in a direction d ∈ Rn is defined by f (x + αd) − f (x) f  (x; d) = lim (3.24) α↓0 α when this limit (finite or infinite) exists, where α ↓ 0 means that α tends to 0 from the positive side (α > 0). For convex f , the limit exists for all d, and f  (x; d) is a convex function in d. For polyhedral convex f , there exists ε > 0, independent of x ∈ dom f , such that f  (x; d) = f (x + d) − f (x)

(||d||1 ≤ ε).

(3.25)

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3.1. Convex Analysis

Y

81

6

Y = f (x)

−f • (p) Y = p, x − f • (p) x Figure 3.1. Conjugate function (Legendre–Fenchel transform).

The conjugate (or convex conjugate) of a function f : Rn → R ∪ {+∞}, where dom f = ∅ is assumed, is a function f • : Rn → R ∪ {+∞} defined by f • (p) = sup{ p, x − f (x) | x ∈ Rn }

(p ∈ Rn ).

(3.26)

This is a convex function in p, being the maximum of (infinitely many) affine functions in p indexed by x. The function f • is also called the (convex) Legendre–Fenchel transform of f , and the mapping f → f • is referred to as the (convex) Legendre– Fenchel transformation. In the favorable situation where f is a smooth convex function and the supremum in (3.26) is attained by a unique x = x(p) for each p, we have f • (p) = p, x(p) − f (x(p)),

(3.27)

where x = x(p) is determined as the solution to the equation ∇f (x) = p. This hints at a geometrical interpretation of the conjugate function. In the case of n = 1 (see Fig. 3.1), for simplicity, the tangent line to the graph Y = f (x) with slope p intersects the Y -axis at a point with the Y -coordinate equal to −f • (p). Similarly, the concave conjugate of a function h : Rn → R ∪ {−∞}, where dom h = ∅, is a function h◦ : Rn → R ∪ {−∞} defined by h◦ (p) = inf{ p, x − h(x) | x ∈ Rn }

(p ∈ Rn ).

(3.28)

Note that h◦ (p) = −(−h)• (−p). Example 3.1. For a convex function ⎧ ⎨ x log x f (x) = 0 ⎩ +∞

(x > 0), (x = 0), (x < 0),

(3.29)

the conjugate is given by f • (p) = exp(p − 1). This can be verified by a simple calculation based on (3.27).

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For a function f , we may think of (f • )• , the conjugate of the conjugate of f , which is called the biconjugate of f and denoted as f •• . The biconjugate of f is the largest closed convex function that is dominated pointwise by f . In particular, the biconjugate δS •• of the indicator function δS of a set S is the indicator function of the closed convex hull of S. Theorem 3.2. The Legendre–Fenchel transform f • is a closed proper convex function for any f with dom f = ∅, and f •• = f for a closed proper convex function f . Hence, the Legendre–Fenchel transformation f → f • gives a symmetric one-to-one correspondence in the class of all closed proper convex functions. As a consequence of Theorem 3.2 and the definition (3.23), we obtain the relationships p ∈ ∂R f (x) ⇐⇒ x ∈ arg min f [−p] ! f (x) + f • (p) = p, x

(3.30) ! x ∈ ∂R f • (p) ⇐⇒ p ∈ arg min f • [−x] for a closed proper convex function f and vectors x, p ∈ Rn . The conjugate δS • of the indicator function δS of a set S ⊆ Rn is expressed as δS • (p) = sup{ p, x | x ∈ S} (p ∈ Rn ), (3.31) which is the support function of S. The support function of a nonempty set is a positively homogeneous closed proper convex function, where a function g, in general, is said to be positively homogeneous if g(λp) = λg(p)

(3.32)

holds for any λ > 0 and p ∈ Rn (this condition yields g(0) = 0 if dom g = ∅). Theorem 3.2 implies a one-to-one correspondence between closed convex sets and positively homogeneous closed proper convex functions. In this sense, positively homogeneous convex functions are convex sets in disguise. For a closed convex function f and a point x in the relative interior of dom f , for example, the directional derivative f  (x; d) is a positively homogeneous closed proper convex function in d and it coincides with the support function of the subdifferential ∂R f (x): f  (x; d) = (δ∂R f (x) )• (d).

(3.33)

The support function of a convex cone S ⊆ Rn agrees with the indicator function of another convex cone, S ∗ = {p ∈ Rn | p, x ≤ 0 (∀ x ∈ S)},

(3.34)

called the polar cone of S. By Theorem 3.2, (S ∗ )∗ = S for a closed convex cone S. When S is represented as S = {x ∈ Rn | ak , x ≤ 0 (k = 1, . . . , m)},

(3.35)

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S1 S2 p∗ k

Figure 3.2. Separation for convex sets.

with ak ∈ Rn (k = 1, . . . , m), we have ( ∗

S =

) * m )  p ∈ R )p = λk ak , λk ∈ R+ (k = 1, . . . , m) . ) n)

(3.36)

k=1

Note 3.3. A bounded polyhedron can be represented in two different ways: as the convex hull of the vertices (vertex-oriented representation) and as the intersection of finitely many half-spaces described by linear inequalities (face-oriented representation). Let S be a bounded polyhedron, S 0 be the set of its vertices, and S=

2 {x | pk , x ≤ bk } k

be a nonredundant representation of S. Then we have bk = δS 0 • (pk ). This shows that the translation between the two representations, S 0 ←→ {(bk , pk )}, can be regarded as a special case of the Legendre–Fenchel transformation. In fact we have already seen this phenomenon in Theorem 1.9, which gives two equivalent characterizations of base polyhedra, one by exchangeability and the other by submodularity. The exchangeability (B-EXC[Z]) is for vertices and the submodularity for faces; ρ(X) in Theorem 1.9 corresponds to bk in the present notation (and χX = pk ). Recall that we formulated this as the Legendre–Fenchel conjugacy in Theorem 1.10.

The duality principle constitutes the core of convex analysis. It can be stated in many different forms, but we focus here on separation theorems (for sets and for functions) and the Fenchel duality theorem. The following is the separation theorem for convex sets (see Fig. 3.2). Theorem 3.4 (Separation for convex sets). Let S1 , S2 ⊆ Rn be nonempty convex sets.

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Y = α∗ + p∗ , x

Y 6 Y = h(x) x Figure 3.3. Separation for convex and concave functions.

(1) If S1 ∩ S2 = ∅, there exists a nonzero vector p∗ ∈ Rn such that inf{ p∗ , x | x ∈ S1 } ≥ sup{ p∗ , x | x ∈ S2 }.

(3.37)

If, in addition, S1 and S2 are closed and at least one of them is bounded, then the inequality ≥ above can be replaced with strict inequality >. (2) ri S1 ∩ ri S2 = ∅ if and only if there exists a vector p∗ ∈ Rn such that (3.37) holds and sup{ p∗ , x | x ∈ S1 } > inf{ p∗ , x | x ∈ S2 }. (3.38) (3) If S1 is polyhedral, S1 ∩ ri S2 = ∅ if and only if there exists p∗ ∈ Rn such that (3.37) holds and inf{ p∗ , x | x ∈ S1 } > inf{ p∗ , x | x ∈ S2 }.

(3.39)

Proof. See Theorems 11.3 and 20.2 and Corollary 11.4.2 of Rockafellar [176]. The separation theorem for convex functions, illustrated in Fig. 3.3, asserts the existence of an affine function that lies between a convex function and a concave function. Theorem 3.5 (Separation for convex functions). Let f : Rn → R ∪ {+∞} be a proper convex function and h : Rn → R ∪ {−∞} a proper concave function, and assume that (a1) or (a2) below is satisfied: (a1) ri (dom f ) ∩ ri (dom h) = ∅, (a2) f and h are polyhedral and dom f ∩ dom h = ∅. If f (x) ≥ h(x) (∀ x ∈ Rn ), there exist α∗ ∈ R and p∗ ∈ Rn such that f (x) ≥ α∗ + p∗ , x ≥ h(x)

(∀ x ∈ Rn ).

(3.40)

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Proof. The proof is based on Theorem 3.4 applied to epigraphs. See Corollary 5.1.6 in Stoer–Witzgall [194] and the proof of Theorem 31.1 in Rockafellar [176]. Another expression of the duality principle is in the form of the Fenchel duality. This is a min-max relation between a pair of convex function f and concave function h and their conjugate functions f • and h◦ . We include a proof to demonstrate the equivalence of the Fenchel duality and the separation for functions. Theorem 3.6 (Fenchel duality). Let f : Rn → R ∪ {+∞} be a proper convex function and h : Rn → R ∪ {−∞} a proper concave function, and assume that at least one of the following four conditions (a1)–(b2) is satisfied: (a1) ri (dom f ) ∩ ri (dom h) = ∅, (a2) f and h are polyhedral, and dom f ∩ dom h = ∅, (b1) f and h are closed 37 , and ri (dom f • ) ∩ ri (dom h◦ ) = ∅, (b2) f and h are polyhedral, and dom f • ∩ dom h◦ = ∅. Then it holds that inf{f (x) − h(x) | x ∈ Rn } = sup{h◦ (p) − f • (p) | p ∈ Rn }.

(3.41)

Moreover, if this common value is finite, the supremum is attained by some p ∈ dom f • ∩ dom h◦ under the assumption of (a1) or (a2), and the infimum is attained by some x ∈ dom f ∩ dom h under the assumption of (b1) or (b2). Proof. By the definitions (3.26) and (3.28) of the conjugate functions, we have f • (p) ≥ p, x − f (x),

h◦ (p) ≤ p, x − h(x)

for any x and p. This shows inf ≥ sup in (3.41). Hence (3.41) holds if inf = −∞ or sup = +∞. In what follows we assume inf > −∞ and sup < +∞. Suppose that (a1) or (a2) is satisfied. Then inf in (3.41) is of finite value, say, Δ, and Theorem 3.5 applies to (f − Δ, h) and yields some α∗ ∈ R and p∗ ∈ Rn such that (∀ x ∈ Rn ). f (x) − Δ ≥ α∗ + p∗ , x ≥ h(x) This means that f • (p∗ ) ≤ −α∗ −Δ and h◦ (p∗ ) ≥ −α∗ , implying inf = Δ ≤ h◦ (p∗ )− f • (p∗ ) ≤ sup and hence (3.41). This also shows that p∗ attains the supremum. In the remaining case where (b1) or (b2) is satisfied, we can use a similar argument for (f • , h◦ ) on the basis of the identities (f • )• = f and (h◦ )◦ = h shown in Theorem 3.2. In the case (b2) note that the conjugate function of a polyhedral convex function is again polyhedral. We note that, if the supremum in (3.41) is attained by p = p∗ , then arg min(f − h) = arg min f [−p∗ ] ∩ arg max h[−p∗ ]. 37 By

this we mean that f and −h are closed convex functions.

(3.42)

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Example 3.7. The separation theorem and trated for the convex function ⎧ ⎨ x log x 0 f (x) = ⎩ +∞

the Fenchel duality theorem are illus(x > 0), (x = 0), (x < 0)

and the concave function h(x) = −f (−x). The graphs of Y = f (x) and Y = h(x) are tangent to the Y -axis at the origin (0, 0), and therefore there exists no separating affine function, although f (x) ≥ h(x) (∀ x). This does not contradict Theorem 3.5, since neither (a1) nor (a2) is satisfied. This shows the importance of the conditions (a1) and (a2) in Theorem 3.5. The conjugate functions are given by f • (p) = exp(p − 1) and h◦ (p) = − exp(p − 1), and hence  0 (x = 0), f (x) − h(x) = h◦ (p) − f • (p) = −2 exp(p − 1). +∞ (x = 0), Therefore, the infimum and the supremum in the Fenchel duality (3.41) are both equal to 0; the infimum is attained by x = 0, whereas the supremum is not attained. Note that the condition (b1) in Theorem 3.6 is met. Example 3.8. The infimum and the supremum in the Fenchel duality (3.41) can be distinct if none of the conditions (a1)–(b2) in Theorem 3.6 is satisfied. For the convex function f and the concave function h in x = (x(1), x(2)) defined by  0 (x(1) = 0, x(2) ≥ 0), f (x) = +∞ (otherwise), ⎧ 1 (x(1)x(2) ≥ 1, x(1) > 0, x(2) > 0), ⎨ 3 h(x) = x(1)x(2) (x(1)x(2) ≤ 1, x(1) ≥ 0, x(2) ≥ 0), ⎩ −∞ (otherwise), we have inf = 0 and sup = −1 in (3.41). Note that dom f • = {p | p(2) ≤ 0} and dom h◦ = {p | p(1) ≥ 0, p(2) ≥ 0}, which shows that ri (dom f • ) ∩ ri (dom h◦ ) = ∅, the failure of condition (b1) in Theorem 3.6. The addition (3.19) and the infimal convolution (3.20) are conjugate operations with respect to the Legendre–Fenchel transformation. For proper convex functions f1 and f2 we have (f1 2 f2 )• = f1 • + f2 • , •





(f1 + f2 ) = f1 2 f2 ,

(3.43) (3.44)

where the latter is true under the assumption that ri (dom f1 ) ∩ ri (dom f2 ) = ∅.

3.2

Linear Programming

Linear programming is, undoubtedly, the most important subclass of convex optimization problems. Some fundamental facts about duality and integrality in linear programming are described here.

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87

We start with a fundamental fact about linear inequality systems, known as the Farkas lemma, with a proof based on the separation theorem for convex sets. Theorem 3.9 (Farkas lemma). For a matrix A and a vector b, the conditions (a) and (b) below are equivalent: 38 (a) Ax = b for some nonnegative x ≥ 0. (b) y  b ≥ 0 for any y such that y  A ≥ 0 . Proof. [(a) ⇒ (b)]: It follows from Ax = b, x ≥ 0, and y  A ≥ 0 that y  b = y  Ax ≥ 0. [(b) ⇒ (a)]: Let S be the n convex cone generated by the column vectors aj (j = 1, . . . , n) of A; i.e., S = { j=1 x(j)aj | x(j) ≥ 0}. If (a) fails, then b ∈ / S, which implies, by the separation theorem for convex sets (Theorem 3.4), that y  aj ≥ 0 (j = 1, . . . , n) and y  b < 0 for some y. A linear programming problem is an optimization problem to minimize or maximize a linear objective function subject to linear equality/inequality constraints. Such a problem is also termed a linear program, often abbreviated to LP. Given an m × n matrix A, an m-dimensional vector b, and an n-dimensional vector c, it is convenient to consider a pair of LPs: [Primal problem] Minimize c x subject to Ax = b, x ≥ 0.

[Dual problem] Maximize b y subject to A y ≤ c.

(3.45)

The LPs in such a pair are said to be dual to each other. For convenience of reference, we call the problem on the left the primal problem and the one on the right the dual problem. We denote the feasible regions of the above problems by P = {x ∈ Rn | Ax = b, x ≥ 0},

D = {y ∈ Rm | A y ≤ c}.

The linear programming duality is stated in the following theorem. Theorem 3.10 (LP duality). (1) [Weak duality] c x ≥ b y for any x ∈ P and y ∈ D. (2) [Strong duality] If P = ∅ or D = ∅, then39 inf{c x | x ∈ P } = sup{b y | y ∈ D}.

(3.46)

This common value is finite if and only if both P and D are nonempty, and in that case, the infimum and the supremum are attained by some x ∈ P and y ∈ D, respectively. 38 Inequality between vectors means componentwise inequality; e.g., x ≥ 0 for x = (x(j))n j=1 means x(j) ≥ 0 for j = 1, . . . , n. 39 By convention, inf x∈P = +∞ if P = ∅ and supy∈D = −∞ if D = ∅.

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(3) [Complementarity] Assume x ∈ P and y ∈ D. Then x is optimal in the primal problem and y is optimal in the dual problem if and only if x(j) = 0 or (A y − c)(j) = 0 for each j = 1, . . . , n, 

(3.47)



where (A y − c)(j) denotes the jth component of A y − c. Proof. (1) is easy to see. The essence of this theorem lies in (2), which can be derived from the Farkas lemma. Then (3) follows. See, e.g., Chv´atal [24], Dantzig [36], Schrijver [181], and Vanderbei [206]. Linear programming acquires combinatorial flavor through integrality considerations. An LP described by integer data (an integer matrix A and integer vectors b and c) may or may not have an integer optimal solution. The major interest in this context is under which condition an integer optimal solution is guaranteed. An integer matrix is totally unimodular if every minor is equal to ±1 or 0. Each entry of a totally unimodular matrix is either ±1 or 0. Example 3.11. The incidence matrix of a graph is a typical example of a totally unimodular matrix. Let G = (V, E) be a directed graph with vertex set V and arc set E, where we assume no self-loops exist. The incidence matrix of G, say, A, is a matrix such that the row set is indexed by V and the column set by E, and the (v, a)-entry is given by ⎧ ⎨ +1 (v is the initial vertex of arc a), −1 (v is the terminal vertex of arc a), (v, a)-entry of A = ⎩ 0 (otherwise). An example of an incidence matrix is (2.12). ⊂ ⊂ Example 3.12. Let V be a finite set. For a chain C : X1 ⊂

= X2 = · · · = Xm of subsets of V , the incidence matrix of C is an m × |V | matrix C defined by  1 (j ∈ Xi ) Cij = (1 ≤ i ≤ m, j ∈ V ). 0 (j ∈ / Xi )

Note that the ith row of C is the characteristic vector of Xi . For two chains C 1 and 1 C 2 , with incidence matrices C 1 and C 2 , the matrix A = [ C C 2 ] is totally unimodular. To prove this, it suffices to assume that A is square and to show det A ∈ ⊂ k⊂ k {0, ±1}. Let C k : X1k ⊂

= X2 = · · · = Xmk (k = 1, 2) be the chains and, for k = 1, 2, define Dk to be the matrix with the ith row of C k replaced with the characteristic D1 k for i = 1, . . . , mk , where X0k = ∅. Put A˜ = [ −D Then vector of Xik \ Xi−1 2 ]. ˜ ˜ ˜ det A = ± det A by the construction. We also have det A ∈ {0, ±1}, since A, having at most one entry of 1 and at most one entry of −1 in each column, can be regarded as a submatrix of the incidence matrix of a graph (see Example 3.11). The following theorem relates the total unimodularity of the coefficient matrix to the integrality of optimal solutions of LPs.

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Theorem 3.13. Let A be a totally unimodular matrix. (1) If b is integral, the primal LP in (3.45) has an integral optimal solution x ∈ Zn as long as it has an optimal solution. (2) If c is integral, the dual LP in (3.45) has an integral optimal solution y ∈ Zm as long as it has an optimal solution. Proof. See, e.g., Chv´ atal [24], Cook–Cunningham–Pulleyblank–Schrijver [26], Korte– Vygen [115], Lawler [119], and Schrijver [181]. Such a theorem enables us to treat combinatorial problems via linear programming. Let us demonstrate this for the weighted bipartite matching problem. Let G = (V + , V − ; E) be a bipartite graph with vertex bipartition (V + , V − ) and arc set E. A set M of arcs of G is called a matching if each vertex of G is incident to at most one arc of M and a perfect matching if each vertex of G is incident to exactly one arc of M . We have |M | = |V + | = |V − | for a perfect matching M . Proposition 3.14. Let G = (V + , V − ; E) be a bipartite graph with a perfect matching, and let c : V + × V − → R ∪ {+∞} be a (weight or cost) vector such that c(u, v) < +∞ ⇔ (u, v) ∈ E. Then there exist a vector 40 pˆ : V + ∪ V − → R and orderings of vertices V + = {u1 , . . . , um } and V − = {v1 , . . . , vm } such that  c(ui , vj ) + pˆ(ui ) − pˆ(vj )

= 0 (1 ≤ i = j ≤ m), ≥ 0 (1 ≤ i, j ≤ m).

(3.48)

The set of arcs {(ui , vi ) | i = 1, . . . , m} is a perfect matching of minimum weight, and, therefore, Minimum weight of a perfect matching =

m 

(ˆ p(vi ) − pˆ(ui )) .

(3.49)

i=1

Proof. Consider the primal LP in (3.45) in which A is the incidence matrix of G with arcs directed from V + to V − , b is an integer vector defined by  b(v) =

1 (v ∈ V + ), −1 (v ∈ V − ),

and c is the vector of weights (c(u, v) | (u, v) ∈ E). Since A is totally unimodular by Example 3.11, the optimal solution x may be chosen, by Theorem 3.13, to be an integer vector, which, being a {0, 1}-vector because of the constraints, can be interpreted as the incidence vector of an optimal matching. The dual optimal solution can be identified with a vector pˆ : V + ∪ V − → R, and the condition (3.48) follows from the dual feasibility and the complementarity. 40 This

pˆ is called a potential or an optimal potential .

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Chapter 3. Convex Analysis, Linear Programming, and Integrality

Integrality for a Pair of Integral Polyhedra

Discrete duality often boils down to integrality for a pair of integral polyhedra. In this section we observe some fundamental facts about the intersection and the Minkowski sum of a pair of integral polyhedra. In so doing we intend to gain a better understanding of the subtlety in the relationship between the integrality of polyhedra and the convexity of discrete sets. A polyhedron is said to be rational if it is described by a finite system of linear inequalities with rational coefficients, i.e., if all the coefficients aij and bi in (3.10) can be chosen to be rational numbers. A rational polyhedron P ⊆ Rn is an integral polyhedron if P = P ∩ Zn , i.e., if it coincides with the convex hull (convex closure) of the integer points contained in it. Let us say that a discrete set S ⊆ Zn is hole free if S = S ∩ Zn ,

(3.50)

which means that all the integer points contained in the convex hull of S belong to S itself. A finite set of integer points is hole free if and only if it is the set of integer points in some integral polytope.41 The hole-free property (3.50) seems to be a natural requirement for a discrete set to be qualified as being convex. This is indeed compatible with our previous naive idea of convexity for discrete functions in terms of the extensibility to convex functions formulated in (1.11). We define the indicator function δS : Zn → {0, +∞} of a discrete set S by  0 (x ∈ S), (3.51) δS (x) = +∞ (x ∈ / S). Then a discrete set is hole free if and only if its indicator function is extensible to a convex function. We are now interested in the compatibility of the hole-free property with the Minkowski addition (3.21), which is one of the fundamental operations in convex analysis. We define the Minkowski sum S1 + S2 of two discrete sets S1 , S2 ⊆ Zn by S1 + S2 = {x1 + x2 | x1 ∈ S1 , x2 ∈ S2 },

(3.52)

which we also call the discrete Minkowski sum or the integral Minkowski sum to emphasize the discreteness. If the hole-free property can be qualified as a discrete version of convexity, this property should be preserved in Minkowski addition. Contrary to this optimistic expectation, the Minkowski sum of hole-free sets can have a hole, as is demonstrated in Example 3.15 below. Example 3.15. Two sets S1 = {(0, 0), (1, 1)},

S2 = {(1, 0), (0, 1)}

are hole free with S1 = S1 ∩ Z2 and S2 = S2 ∩ Z2 (see Fig. 3.4). Nevertheless, the discrete Minkowski sum S1 + S2 = {(1, 0), (0, 1), (2, 1), (1, 2)} 41 A

polytope is a bounded polyhedron.

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2

6

S1 2

1

6

91

S2 2

1 -

0 0

1

1 -

0

2

6 S1 + S2

0

1

2

-

0 0

1

2

Figure 3.4. Nonconvexity in Minkowski sum.

has a hole at (1, 1) and, therefore, S1 + S2 = S1 + S2 ∩ Z2 . We observe, in passing, that S1 ∩ S2 = {(1/2, 1/2)}, S1 ∩ S2 = ∅, which shows S1 ∩ S2 = S1 ∩ S2 . The above example issues the following warnings to us about the integrality for a pair of hole-free discrete sets S1 and S2 with Sk = Sk ∩ Zn (k = 1, 2). 1. [S1 + S2 = S1 + S2 ∩ Zn ] is not always true. 2. [S1 ∩ S2 = ∅ ⇒ S1 ∩ S2 = ∅] is not always true. 3. [S1 ∩ S2 = S1 ∩ S2 ] is not always true. 4. The intersection P1 ∩ P2 of integral polyhedra Pk ⊆ Rn (k = 1, 2) is not always an integral polyhedron. The above facts suggest that the hole-free property (3.50) alone is not appropriate as the condition of discrete convexity for sets. Some deeper combinatorial properties are needed. The first two properties above will turn out to be critical in many situations, and, in fact, they are essentially equivalent to each other. Proposition 3.16. Suppose that a family F of sets of integer points has the property (3.53) S ∈ F , x ∈ Zn =⇒ S = S ∩ Zn , x − S ∈ F , where x − S = {x − y | y ∈ S}. Then conditions (a) and (b) below are equivalent for F . (a) ∀ S1 , S2 ∈ F : S1 ∩ S2 = ∅ =⇒ S1 ∩ S2 = ∅. (b) ∀ S1 , S2 ∈ F : S1 + S2 = S1 + S2 ∩ Zn . Proof. (a) ⇒ (b): For x ∈ S1 + S2 ∩ Zn we have x ∈ (S1 + S2 ) ∩ Zn by Proposition 3.17 (4) below. Hence S1 ∩ S2 = ∅ for S1 = S1 and S2 = x − S2 . By (a), there exists y ∈ S1 ∩ S2 . Then y ∈ S1 and ∃ z ∈ S2 : y = x − z. Therefore, x ∈ S1 + S2 .

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(b) ⇒ (a): Suppose S1 ∩ S2 = ∅ and put S1 = S1 and S2 = −S2 . Then 0 ∈ + S2 = S1 + S2 (see Proposition 3.17 (4) below). By (b) we obtain 0 ∈ S1 + S2 , which is equivalent to S1 ∩ S2 = ∅.

S1

We say that a family F of sets of integer points has convexity in intersection if (a) above is true and convexity in Minkowski sum if (b) is true. It will be shown in sections 4.6 and 5.4 that the families of M-convex sets and L-convex sets, respectively, have these properties. Finally, we mention basic relations that are always true. Proposition 3.17. Assume Sk = Sk ∩ Zn for k = 1, 2. (1) S1 ∩ S2 ⊇ S1 ∩ S2 . (2) S1 ∩ S2 = S1 ∩ S2 ∩ Zn . (3) S1 + S2 ⊆ S1 + S2 ∩ Zn . (4) S1 + S2 = S1 + S2 . Proof. (1), (2), and (3) are obvious. We prove (4). First, note that S1 + S2 ⊇ S1 + S2 , which follows from S1 + S2 ⊇ S1 + S2 and the convexity of S1 + S2 . To show the reverse inclusion, take   x= λi yi + μj z j ∈ S 1 + S 2 , i



j

where λi ≥ 0, i λi = 1, yi ∈ S1 , μj ≥ 0, j μj = 1, and zj ∈ S2 (the summations being finite sums). With νij = λi μj we obtain   νij (yi + zj ), νij ≥ 0, νij = 1, x= i,j

i,j

which shows x ∈ S1 + S2 .

3.4

Integrally Convex Functions

Integrally convex functions form a fairly general class of discrete convex functions, for which global optimality is guaranteed by local optimality (in an appropriate sense). Almost all discrete convex functions treated in this book, including L-convex and M-convex functions, fall into this category. For two integer vectors, a, b ∈ (Z∪{±∞})n , the integer interval [a, b] = [a, b]Z is defined by [a, b] = [a, b]Z = {x ∈ Zn | a(i) ≤ x(i) ≤ b(i) (i = 1, . . . , n)},

(3.54)

where, if a(i) = −∞, for example, a(i) ≤ x(i) is to be understood as −∞ < x(i). The restriction of a function f : Zn → R ∪ {+∞} to an interval [a, b] is defined as the function f[a,b] : Zn → R ∪ {+∞} given by  f (x) (x ∈ [a, b]), f[a,b] (x) = (3.55) +∞ (x ∈ / [a, b]).

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Let f : Zn → R ∪ {+∞} be a function defined on the integer lattice, where it is a tacit agreement that the effective domain domZ f is nonempty. The convex closure of f is defined to be a function f : Rn → R ∪ {±∞} given by f (x) =

sup

p∈Rn ,α∈R

{ p, x + α | p, y + α ≤ f (y) (∀ y ∈ Zn )}

(x ∈ Rn ).

(3.56)

If this function f coincides with f on integer points, i.e., if (x ∈ Zn ),

f (x) = f (x)

(3.57)

we say that f is convex extensible and call f the convex extension of f .42 The following fact is easy to see. Proposition 3.18. If a function f : Zn → R ∪ {+∞} is convex extensible, then arg min f [−p] is hole free for each p ∈ Rn . The converse is also true if domZ f is bounded. A local version of the convex extension of a function f can be defined by relaxing the requirement in the definition (3.56) of the convex closure. Instead of imposing the inequality p, y + α ≤ f (y) for all y ∈ Zn , we ask for this condition only for points y ∈ Zn lying in a neighborhood of x ∈ Rn . To be specific, we define the integral neighborhood of x ∈ Rn (see Fig. 3.5) by N (x) = {y ∈ Zn | x(i) ≤ y(i) ≤ x(i) (1 ≤ i ≤ n)}

(x ∈ Rn ),

(3.58)

where, for z ∈ R in general, z denotes the smallest integer not smaller than z (rounding up to the nearest integer) and z the largest integer not larger than z (rounding down to the nearest integer). Note an alternative expression N (x) = {y ∈ Zn | ||x − y||∞ < 1} using the ∞ -norm

||z||∞ = max |z(i)| 1≤i≤n

(x ∈ Rn )

(3.59)

(z ∈ Rn ).

(3.60)

With this neighborhood we define the local convex extension of f by f˜(x) =

sup

p∈Rn ,α∈R

{ p, x + α | p, y + α ≤ f (y) (∀ y ∈ N (x))}

(x ∈ Rn ). (3.61)

Note the obvious relations f˜(x) ≥ f (x)

(x ∈ Rn ),

f˜(x) = f (x) (x ∈ Zn ).

We have an alternative expression ⎧ ) ⎫ ) ⎨  ⎬ )  f˜(x) = inf λy f (y))) λy y = x, (λy )y∈N (x) ∈ Λ ⎩ ⎭ ) y∈N (x)

(3.62)

(x ∈ Rn ), (3.63)

y∈N (x)

42 We say that f is concave extensible if −f is convex extensible and then −(−f ) is the concave extension of f .

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x

x

Figure 3.5. Integral neighborhood N (x) of x (◦: point of N (x)).

with

⎧ ⎨

⎫ ) ) ⎬ )  Λ = (λy )y∈N (x) )) λy = 1, λy ≥ 0 (∀ y ∈ N (x)) , ⎩ ⎭ )y∈N (x)

as a consequence of LP duality (Theorem 3.10). In the univariate case (n = 1), the graph of f˜ consists of line segments connecting the points {(z, f (z)) | z ∈ Z} in the natural order. The local convex extension f˜ is convex on every unit interval [z, z + 1]R = {x ∈ Rn | z(i) ≤ x(i) ≤ z(i) + 1 (1 ≤ i ≤ n)} with an integral point z ∈ Zn , but is not necessarily convex in the entire space Rn . If f˜ is convex on Rn , the function f is said to be integrally convex . Alternatively, we can define f is integrally convex ⇐⇒ f˜(x) = f (x)

(x ∈ Rn ).

(3.64)

In particular, an integrally convex function is convex extensible. A function h is called integrally concave if −h is integrally convex. Note the following fact. Proposition 3.19. For a function f : Zn → R ∪ {+∞}, f is integrally convex ⇐⇒ f[a,b] is integrally convex for any a, b ∈ Zn . Example 3.20. Here is an example of a convex-extensible function that is not integrally convex. Let f : Z2 → R be defined by f (x) = |x(1) − 2x(2)| for x = (x(1), x(2)) ∈ Z2 . Obviously, this function is extensible to a convex function f (x) = |x(1) − 2x(2)| defined for x = (x(1), x(2)) ∈ R2 . In particular, we have f (1, 1/2) = 0. On the other hand, we have f˜(1, 1/2) = 1 since N (x) = {(1, 0), (1, 1)} for x = (1, 1/2) and f (1, 0) = f (1, 1) = 1. Hence f (1, 1/2) = f˜(1, 1/2), which shows that f is not integrally convex. The global minimum of an integrally convex function can be characterized by a local optimality. This is the key property of integrally convex functions that justifies this notion is as follows. Theorem 3.21. For an integrally convex function f : Zn → R ∪ {+∞} and x ∈ domZ f , we have f (x) ≤ f (y) (∀ y ∈ Zn ) ⇐⇒ f (x) ≤ f (x+χY −χZ ) (∀ Y, Z ⊆ {1, . . . , n}). (3.65)

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95

Proof. It suffices to show ⇐. Put N1 (x) = {y ∈ Rn | ||y − x||∞ ≤ 1} for x ∈ domZ f . By (3.63) and (3.65) we have f (x) ≤ f˜(y) for all y ∈ N1 (x). Combining this with integral convexity (3.64) shows f (x) ≤ f (y) (∀ y ∈ N1 (x)), the local minimality of f at x. Since f is convex, x is a global minimizer of f by Theorem 1.1 and, a fortiori, a global minimizer of f . The following variant of the above theorem will be used later. Proposition 3.22. Let f : Zn → R ∪ {+∞} be an integrally convex function such that f (z + 1) = f (z) for all z ∈ Zn . For x ∈ domZ f we have f (x) ≤ f (y) (∀ y ∈ Zn ) ⇐⇒ f (x) ≤ f (x + χY ) (∀ Y ⊆ {1, . . . , n}).

(3.66)

Proof. It suffices to show that the latter condition in (3.66) implies f (x) ≤ f (x + χY − χZ ) for any disjoint Y and Z. On putting U = {1, . . . , n} \ (Y ∪ Z) and x◦ = x − 1 we have f (x) = f (x◦ ) and   1 f (x + χY − χZ ) = f (x◦ + χU + 2χY ) ≥ 2f x◦ + χU + χY − f (x◦ ). 2 Here we have f (x◦ + 12 χU +χY ) ≥ f (x◦ ), since f (x◦ + 12 χU +χY ) can be represented, by integral convexity, as a convex combination of f (x◦ + χW + χY ) with W ⊆ U and f (x◦ + χW + χY ) ≥ f (x◦ ) for any W ⊆ U by the assumption.

Note 3.23. The optimality criterion in Theorem 3.21 is certainly local, but not satisfactory from the computational complexity viewpoint. We need O(3n ) function evaluations to verify the local optimality condition in (3.65). A function f : Zn → R ∪ {+∞} is called a separable convex function if it can be represented as f (x) =

n 

fi (x(i))

(x = (x(i))ni=1 ∈ Zn ),

(3.67)

i=1

with univariate discrete convex functions fi ∈ C[Z → R] (i = 1, . . . , n), where C[Z → R] = {ϕ : Z → R ∪ {+∞} | domZ ϕ = ∅, ϕ(t − 1) + ϕ(t + 1) ≥ 2ϕ(t) (t ∈ Z)}

(3.68)

is the set of univariate discrete convex functions. Similarly, we denote by C[Z → Z] the set of integer-valued univariate discrete convex functions. Proposition 3.24. The sum of an integrally convex function and a separable convex function is an integrally convex function.

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f0 is integrally convex and fi ∈ Proof. Put f (x) = f0 (x) + ni=1 fi (x(i)), where C[Z → R] for i = 1, . . . , n. For any (λy ) ∈ Λ with y∈N (x) λy y = x, we have  y∈N (x)

λy

n 

fi (y(i)) =

i=1

n  

λy fi (y(i)) =

i=1 y∈N (x)

n 

f˜i (x(i)).

i=1

It follows from this and (3.63) that f˜(x) = f˜0 (x) +

n 

f˜i (x(i)) = f0 (x) +

n 

i=1

fi (x(i)),

i=1

which shows the convexity of f˜. The following proposition is an immediate corollary of Proposition 3.24, where, for p ∈ Rn , we define f [−p](x) = f (x) − p, x

(x ∈ Zn ),

(3.69)

arg min f [−p] = {x ∈ Z | f [−p](x) ≤ f [−p](y) (∀ y ∈ Z )}. n

n

(3.70)

Proposition 3.25. (1) A separable convex function is integrally convex. (2) f [−p] is integrally convex for integrally convex f and vector p ∈ Rn . A set of integer points S ⊆ Zn is said to be integrally convex if its indicator function δS is an integrally convex function. This means that a set S is integrally convex if and only if (3.71) x ∈ S =⇒ x ∈ S ∩ N (x) for any x ∈ Rn . We also have S is an integrally convex set ⇐⇒ S ∩ N (x) = S ∩ N (x)

(∀ x ∈ Rn )

(3.72)

(see Fig. 3.6). An integrally convex set is hole free. Proposition 3.26. S = S ∩ Zn for an integrally convex set S. Proof. This follows from (3.72), since N (x) = {x} for an integer point x.

Note 3.27. The family of integrally convex sets has neither convexity in intersection nor convexity in Minkowski sum. Example 3.15 shows this. The integral convexity of a function can be characterized by the integral convexity of the minimizers (Theorem 3.29 below).

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Integrally convex

97

Not integrally convex

Not integrally convex

Figure 3.6. Concept of integrally convex sets.

Proposition 3.28. Let f : Zn → R ∪ {+∞} be an integrally convex function. (1) domZ f is an integrally convex set. (2) For each p ∈ Rn , arg min f [−p] is an integrally convex set. Proof. (1) By f = f˜ and the definition of f˜ we have dom f ∩ N (x) = dom f ∩ N (x) = dom f˜ ∩ N (x) = dom f ∩ N (x). Then (3.72) shows the integral convexity of dom f . (2) We assume p = 0 by Proposition 3.25 (2) and use (3.71) for S = arg min f . For x ∈ S we have min f = f (x) = f˜(x) and therefore x ∈ S ∩ N (x). Theorem 3.29. Suppose a function f : Zn → R ∪ {+∞} has a nonempty bounded effective domain. Then f is an integrally convex function ⇐⇒ arg min f [−p] is an integrally convex set for each p ∈ Rn . Proof. The implication ⇒ was shown in Proposition 3.28. For the converse we are / dom f , we have f˜(x) = f (x) = +∞ by to show f˜(x) = f (x) for x ∈ Rn . If x ∈ dom f = dom f and (3.62). Assume x ∈ dom f , and consider a pair of (mutually dual) LPs: (P)

Maximize subject to

(D)

Minimize subject to

p, x + α p, y + α ≤ f (y) (y ∈ dom f ), p ∈ Rn , α ∈ R.  λy f (y) y∈dom f y∈dom f

λy y = x,



λy = 1, λy ≥ 0 (y ∈ dom f ).

y∈dom f

Here (p, α) and (λy | y ∈ dom f ) are the variables of (P) and (D), respectively. Problem (P) is obviously feasible, and so is (D) by x ∈ dom f . Let (p∗ , α∗ ) and λ∗ = (λ∗y | y ∈ dom f ) be optimal solutions of (P) and (D), respectively. Then

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(3.56), (3.62), (3.63), and LP duality (Theorem 3.10 (2)) imply f (x) = p∗ , x + α∗ =



λ∗y f (y) ≤ f˜(x).

(3.73)

y∈dom f

It remains to show that the inequality here is in fact an equality. To denote the set of tight constraints at (p∗ , α∗ ), we put S = {y ∈ dom f | p∗ , y + α∗ = f (y)} = arg min f [−p∗ ](y). y∈dom f

We have {y ∈ dom f | λ∗y > 0} ⊆ S by the complementarity (Theorem 3.10 (3)). Hence x ∈ S, and furthermore, x ∈ S ∩ N (x) by the integral convexity of S and ˜ = (λ ˜ y | y ∈ dom f ) to (3.71). Therefore, there exists another optimal solution λ ˜ y > 0} ⊆ S ∩ N (x). Then, by (3.63), we obtain (D) satisfying {y | λ  y∈dom f

λ∗y f (y) =

 y∈dom f

˜y f (y) = λ



˜ y f (y) ≥ f˜(x), λ

y∈N (x)

which shows that the inequality in (3.73) is an equality. We mention a technical fact to be used in section 8.1. Proposition 3.30. For an integer-valued integrally convex function f : Zn → Z ∪ {+∞} and p ∈ Rn , we have arg min f [−p] = ∅ if inf f [−p] > −∞. Proof. The proof is not difficult; see Lemma 6.13 in Murota–Shioura [152].

Note 3.31. The intersection of integrally convex sets is not necessarily integrally convex; e.g., S1 = {(0, 0, 0), (0, 1, 1), (1, 1, 0), (1, 2, 1)} and S2 = {(0, 0, 0), (0, 1, 0), (1, 1, 1), (1, 2, 1)} are integrally convex, but their intersection S1 ∩ S2 = {(0, 0, 0), (1, 2, 1)} is not, since (S1 ∩ S2 ) ∩ N (x) = ∅ for x = (1/2, 1, 1/2) ∈ S1 ∩ S2 . This implies also that the sum of integrally convex functions is not necessarily integrally convex. The discrete separation theorem (see section 1.2) does not hold for integrally convex functions; Example 1.5 shows this. Note 3.32. A function f : Zn → R ∪ {+∞} is said to be a Miller’s discrete convex function if min{f (z) | z ∈ N (αx + (1 − α)y)} ≤ αf (x) + (1 − α)f (y)

(3.74)

holds for any x, y ∈ domZ f and any α ∈ [0, 1]R . An integrally convex function satisfies this condition. The optimality criterion (3.65) stated for integrally convex functions in Theorem 3.21 is in fact valid for Miller’s discrete convex functions.

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99

Bibliographical Notes The introduction to convex analysis in section 3.1 is kept to the minimum needed for later developments in this book. For a systematic and comprehensive account, see Borwein–Lewis [17], Hiriart-Urruty–Lemar´echal [89], Rockafellar [176], Rockafellar [177], Rockafellar–Wets [179], and Stoer–Witzgall [194]. In particular, see Theorems 11.3 and 20.2 and Corollary 11.4.2 of [176] for separation for convex sets (Theorem 3.4); Corollary 5.1.6 in [194] and the proof of Theorem 31.1 in [176] for separation for convex functions (Theorem 3.5); and Theorem 31.1 in [176] and Corollary 5.1.4 in [194] for Fenchel duality (Theorem 3.6). Example 3.8 is taken from [194]. References on linear programming abound in the literature; see, e.g., Chv´atal [24], Dantzig [36], Schrijver [181], and Vanderbei [206]. Matching is one of the central topics in graph theory, the standard reference being Lov´asz–Plummer [125]. Matching is also fundamental in combinatorial optimization; see Cook–Cunningham– Pulleyblank–Schrijver [26], Du–Pardalos [43], Korte–Vygen [115], Lawler [119], and Nemhauser–Wolsey [167]. Section 3.3 is a collection of basic facts, as presented in Murota [147]. The terms convexity in intersection and convexity in Minkowski sum are coined here. Proposition 3.16 is explicit in Danilov–Koshevoy [32], where a general framework for convexity in intersection and convexity in Minkowski sum is provided. The concept of integrally convex functions was introduced by Favati–Tardella [49], where the effective domains are assumed to be integer intervals. The optimality criterion (Theorem 3.21) is in [49]. Propositions 3.24 and 3.28 are due to Murota– Shioura [153], and Theorem 3.29 (implicit in [153]) is taken from Murota [147]. Miller’s discrete convex functions are introduced by Miller [130] along with the optimality criterion (3.65).

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Chapter 4

M-Convex Sets and Submodular Set Functions

M-convex sets form a class of well-behaved discrete convex sets. They are defined in terms of an exchange axiom and correspond one-to-one to integer-valued submodular set functions. An M-convex set is exactly the same as the integer points contained in the base polyhedron associated with some integral submodular function. This chapter, accordingly, is a systematic presentation of known results in the theory of matroids and submodular functions from the viewpoint of discrete convex analysis.

4.1

Definition

Let V be a finite set, say, V = {1, . . . , n}. A nonempty set of integer points B ⊆ ZV is defined to be an M-convex set if it satisfies the following exchange axiom: (B-EXC[Z]) For x, y ∈ B and u ∈ supp+ (x − y), there exists v ∈ supp− (x − y) such that x − χu + χv ∈ B and y + χu − χv ∈ B. Here supp+ (x − y) and supp− (x − y) are the positive support and the negative support of x − y defined in (1.19) and χu is the characteristic vector of u ∈ V . We denote by M0 [Z] the set of M-convex sets. M-convexity thus defined for a set B ⊆ ZV is equivalent to the M-convexity of the indicator function δB : ZV → {0, +∞} (defined in (3.51)). Namely, B is an M-convex set satisfying (B-EXC[Z]) if and only if δB is an M-convex function satisfying (M-EXC[Z]) introduced in section 1.4.2. Recall that we encountered (B-EXC[Z]) in section 1.3.2 as the exchange property that characterizes the sets of integer points associated with a submodular function (see Theorem 1.9). Hence, an M-convex set is exactly the same as the set of integer points contained in the base polyhedron defined by an integer-valued submodular set function. An immediate consequence of the exchange axiom (B-EXC[Z]) is that an Mconvex set lies on a hyperplane {x ∈ RV | x(V ) = r} for some r ∈ Z, where we use 101

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Chapter 4. M-Convex Sets and Submodular Set Functions

the notation x(X) =



x(v)

(x ∈ RV , X ⊆ V ),

(4.1)

v∈X

||x||1 =



|x(v)|

(x ∈ RV ).

(4.2)

v∈V

Proposition 4.1. For an M-convex set B we have x(V ) = y(V ) for any x, y ∈ B. Proof. The proof is by induction on ||x − y||1 . If ||x − y||1 = 0, we obviously have x(V ) = y(V ). The case of ||x − y||1 = 1 is excluded by (B-EXC[Z]). If ||x − y||1 ≥ 2, (B-EXC[Z]) implies y  ≡ y + χu − χv ∈ B, for which we have y  (V ) = y(V ), ||x − y  ||1 = ||x − y||1 − 2, and also x(V ) = y  (V ) by the induction hypothesis. Since an M-convex set lies on a hyperplane {x ∈ RV | x(V ) = r}, we may equivalently consider the projection of an M-convex set along an arbitrarily chosen coordinate axis. We call the projection of an M-convex set an M -convex set . Whereas M -convex sets are conceptually equivalent to M-convex sets, the class of M -convex sets is strictly larger than that of M-convex sets. The simplest example of an M -convex set that is not M-convex is an integer interval [a, b]Z . We focus on M-convex sets in the development of the theory and deal with M -convex sets in section 4.7.

4.2

Exchange Axioms

There are a number of equivalent variants of the exchange axiom (B-EXC[Z]). Whereas (B-EXC[Z]) requires that both x − χu + χv and y + χu − χv belong to B, (B-EXC+ [Z]) below imposes this only on y + χu − χv . Proposition 4.2. For a set B ⊆ ZV , (B-EXC[Z]) is equivalent to the following: (B-EXC+ [Z]) For x, y ∈ B and u ∈ supp+ (x − y), there exists v ∈ supp− (x − y) such that y + χu − χv ∈ B. Proof. It suffices to show (B-EXC+ [Z]) ⇒ (B-EXC[Z]). First, it is easy to see that (B-EXC+ [Z]) implies the following: (B-EXC−loc [Z]) For x, y ∈ B with ||x − y||1 = 4 and v ∈ supp− (x − y), there exists u ∈ supp+ (x − y) such that y + χu − χv ∈ B. To prove the claim by contradiction, we assume that there exists a pair (x, y) for which (B-EXC[Z]) fails. That is, we assume that the set of such pairs D = {(x, y) | x, y ∈ B, ∃ u∗ ∈ supp+ (x − y), ∀ v ∈ supp− (x − y) : / B or y + χu∗ − χv ∈ / B} x − χu∗ + χv ∈

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103

is nonempty. Take a pair (x, y) ∈ D with minimum ||x − y||1 ; we have ||x − y||1 ≥ 4. Fix u∗ ∈ supp+ (x − y) as above, take any u0 ∈ supp+ (x − y − χu∗ ), and put X = {v ∈ supp− (x − y) | x − χu∗ + χv ∈ B}, Y = {v ∈ supp− (x − y) | y + χu0 − χv ∈ B}, where Y = ∅ by (B-EXC+ [Z]). Take any v0 ∈ Y , where we assume v0 ∈ X ∩ Y if X ∩ Y = ∅. Then y  = y + χu0 − χv0 satisfies y  ∈ B and ||x − y  ||1 = ||x − y||1 − 2. We also have (x, y  ) ∈ D, as shown below, a contradiction to the choice of (x, y). It remains to show (x, y  ) ∈ D. We have u∗ ∈ supp+ (x − y  ) and want to show / B. v ∈ supp− (x − y  ), x − χu∗ + χv ∈ B =⇒ y  + χu∗ − χv ∈ Put y  = y  + χu∗ − χv = y + χu0 + χu∗ − χv0 − χv . Note that y + χu∗ − χv ∈ / B, since (x, y) ∈ D and x − χu∗ + χv ∈ B. If X ∩ Y = ∅, we have y + χu∗ − χv ∈ / B and y + χu∗ − χv0 ∈ / B, and therefore, y  ∈ / B by (B-EXC+ [Z]). If X ∩ Y = ∅, we have / B and y + χu0 − χv ∈ / B, and therefore, y  ∈ / B by (B-EXC−loc [Z]). y + χu∗ − χv ∈ In either case we have (x, y  ) ∈ D. We introduce two other variants: (B-EXCw [Z]) For distinct x, y ∈ B, there exist u ∈ supp+ (x − y) and v ∈ supp− (x − y) such that x − χu + χv ∈ B and y + χu − χv ∈ B. (B-EXC− [Z]) For x, y ∈ B and u ∈ supp+ (x − y), there exists v ∈ supp− (x − y) such that x − χu + χv ∈ B. Theorem 4.3. Conditions (B-EXC[Z]), (B-EXCw [Z]), (B-EXC+ [Z]), and (BEXC− [Z]) are equivalent for a set B ⊆ ZV . Proof. The implication (B-EXC[Z]) ⇒ (B-EXCw [Z]) is obvious, and Proposition 4.2 shows (B-EXC[Z]) ⇔ (B-EXC+ [Z]). We also have (B-EXC[Z]) ⇔ (B-EXC− [Z]), since (B-EXC− [Z]) for B is equivalent to (B-EXC+ [Z]) for −B, and (B-EXC[Z]) for B is equivalent to (B-EXC[Z]) for −B. We show (B-EXCw [Z]) ⇒ (B-EXC− [Z]) by induction on ||x − y||1 . Suppose x, y ∈ B and u ∈ supp+ (x − y). By (B-EXCw [Z]) there exist u1 ∈ supp+ (x − y) and v1 ∈ supp− (x − y) such that x − χu1 + χv1 ∈ B and y  = y + χu1 − χv1 ∈ B. If u1 = u, we are done. Otherwise (u1 = u), we have ||x − y  ||1 = ||x − y||1 − 2 and, by the induction hypothesis, (B-EXC− [Z]) applies to (x, y  ) and u ∈ supp+ (x − y  ). Hence, x − χu + χv ∈ B for some v ∈ supp− (x − y  ) ⊆ supp− (x − y).

4.3

Submodular Functions and Base Polyhedra

We introduce here some fundamental facts about submodular set functions, which turn out to describe the convex hull of M-convex sets. Let ρ : 2V → R ∪ {±∞} be a set function. Its effective domain, denoted as dom ρ, is defined to be the family of subsets at which ρ is finite; i.e., dom ρ = {X ⊆ V | −∞ < ρ(X) < +∞}.

(4.3)

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Chapter 4. M-Convex Sets and Submodular Set Functions

Throughout this book we assume, for a set function ρ in general, that ρ(∅) = 0, ρ(V ) is finite, and either ρ : 2V → R ∪ {+∞} or ρ : 2V → R ∪ {−∞}. The Lov´ asz extension 43 of a set function ρ is a function ρˆ : RV → R ∪ {±∞} defined as follows. Given a vector p ∈ RV , we denote by pˆ1 > pˆ2 > · · · > pˆm the distinct values of its components and put Ui = {v ∈ V | p(v) ≥ pˆi }

(i = 1, . . . , m).

(4.4)

Then we have an identity p=

m−1 

(ˆ pi − pˆi+1 )χUi + pˆm χUm ,

(4.5)

i=1

which is a representation of p as a linear combination of χUi (i = 1, . . . , m). The Lov´ asz extension ρˆ is the linear interpolation on the basis of this representation. Namely, ρˆ is defined by ρˆ(p) =

m−1 

(ˆ pi − pˆi+1 )ρ(Ui ) + pˆm ρ(Um )

(p ∈ RV ),

(4.6)

i=1

with reference to the representation (4.5). The Lov´asz extension ρˆ is a positively homogeneous function that coincides with ρ on {0, 1}V in the sense of ρˆ(χX ) = ρ(X)

(X ⊆ V ).

(4.7)

Since pˆi − pˆi+1 > 0 (1 ≤ i ≤ m − 1) and ρ(Um ) = ρ(V ) is finite on the right-hand side of (4.6), we have p ∈ dom ρˆ ⇐⇒ U1 , U2 , . . . , Um−1 ∈ dom ρ.

(4.8)

A set function ρ : 2V → R ∪ {+∞} is said to be submodular if it satisfies ρ(X) + ρ(Y ) ≥ ρ(X ∪ Y ) + ρ(X ∩ Y )

(X, Y ⊆ V ).

(4.9)

We denote by S[R] the set of submodular set functions ρ with ρ(∅) = 0 and ρ(V ) < +∞ and by S[Z] the set of integer valued such submodular set functions; i.e., S[R] = {ρ : 2V → R ∪ {+∞} | ρ is submodular, ρ(∅) = 0, ρ(V ) < +∞}, S[Z] = {ρ : 2V → Z ∪ {+∞} | ρ is submodular, ρ(∅) = 0, ρ(V ) < +∞}.

(4.10) (4.11)

For ρ ∈ S[R], the effective domain D = dom ρ forms a ring family (sublattice of the Boolean lattice 2V ), which means, by definition, that X, Y ∈ D =⇒ X ∪ Y, X ∩ Y ∈ D.

(4.12)

43 The Lov´ asz extension is also called the Choquet integral or the linear extension. Often ρˆ(p) is defined only for nonnegative p, although it is more convenient for us to define it over the entire space RV .

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105

We call a set function μ : 2V → R ∪ {−∞} supermodular if −μ is submodular. The conditions μ(∅) = 0 and μ(V ) > −∞ are always assumed for a supermodular function μ (i.e., −μ ∈ S[R]). For a submodular function ρ ∈ S[R] we consider a polyhedron B(ρ) = {x ∈ RV | x(X) ≤ ρ(X) (∀ X ⊂ V ), x(V ) = ρ(V )},

(4.13)

the base polyhedron associated with ρ. A point (element) of B(ρ) is called a base and an extreme point of B(ρ) is an extreme base. Proposition 4.4. B(ρ) is nonempty for ρ ∈ S[R]. Proof. For simplicity of description we assume that dom ρ has a maximal chain of length n = |V |. On suitably indexing the elements of V , V = {v1 , v2 , . . . , vn }, we have Vj ≡ {v1 , v2 , . . . , vj } ∈ dom ρ for j = 1, . . . , n. Define a vector x ∈ RV by x(vj ) = ρ(Vj ) − ρ(Vj−1 ) for j = 1, . . . , n with V0 = ∅. We show x(X) ≤ ρ(X) by induction on |X|. When |X| = 0, this is obviously true by ρ(∅) = 0. When |X| ≥ 1, let j be the maximum index such that vj ∈ X. Then we have ρ(X) ≥ ρ(X ∩ Vj−1 ) + ρ(X ∪ Vj−1 ) − ρ(Vj−1 ) = ρ(X − vj ) + ρ(Vj ) − ρ(Vj−1 ) ≥ x(X − vj ) + x(vj ) = x(X). Hence follows B(ρ) = ∅. The support function of B(ρ) coincides with the Lov´asz extension of ρ. Proposition 4.5. For a submodular set function ρ ∈ S[R], we have sup{ p, x | x ∈ B(ρ)} = ρˆ(p)

(p ∈ RV ),

(4.14)

where ρˆ is the Lov´ asz extension (4.6) of ρ. Proof. For simplicity we assume that dom ρ has a maximal chain of length n = |V |. Consider a pair of linear programs (LPs): (A)

Maximize subject to

(B)

Minimize subject to

p, x

χX , x ≤ ρ(X) (X ∈ dom ρ \ {V }), χV , x = ρ(V ). {yX ρ(X) | X ∈ dom ρ} {yX | v ∈ X ∈ dom ρ} = p(v) (v ∈ V ), yX ≥ 0 (X ∈ dom ρ \ {V }).

Here x ∈ RV and y = (yX | X ∈ dom ρ) are the variables of (A) and (B), respectively, and p ∈ RV is regarded as a parameter. Note that the equality constraints in (B) can be written as  yX χX = p. (4.15) X∈dom ρ

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Problem (A) is feasible by B(ρ) = ∅ in Proposition 4.4. By LP duality (Theorem 3.10 (2)), the optimal value of (A), max (A), is equal to the optimal value of (B), min (B). Obviously, max (A) is equal to the left-hand side of (4.14). Thus, LHS of (4.14) = max (A) = min (B).

(4.16)

For the feasibility of (B) we have the following statement, where Ui (i = 1, . . . , m) are the subsets determined from p by (4.4). Claim 1: (B) is feasible ⇐⇒ Ui ∈ dom ρ (i = 1, . . . , m). (Proof of Claim 1) For the proof of ⇐, take a maximal chain {Vj } of dom ρ such that {Ui | i = 1, . . . , m} ⊆ {Vj | j = 1, . . . , n}, where we put Vj = {v1 , v2 , . . . , vj } (j = 1, . . . , n). Then the vector y ∗ defined by ⎧ (X = V ), ⎨ p(vn ) ∗ p(vj ) − p(vj+1 ) (X = Vj , 1 ≤ j ≤ n − 1), yX = (4.17) ⎩ 0 (otherwise) is a feasible solution for (B). For the proof of ⇒, take a feasible y that maximizes Γ = {yX |X|2 | X ∈ dom ρ}. Claim 2: C = {X ∈ dom ρ \ {V } | yX > 0} ∪ {V } forms a chain. (Proof of Claim 2) For Y, Z ∈ C \ {V } with yY ≥ yZ > 0, we have the identity yY χY + yZ χZ = yZ χY ∩Z + (yY − yZ )χY + yZ χY ∪Z , where Y ∩ Z, Y ∪ Z ∈ dom ρ. The maximality of Γ implies that |Y \ Z| = 0 or |Z \ Y | = 0, since otherwise Γ would be increased by 2yZ |Y \ Z| |Z \ Y | when the feasible solution is modified according to the above identity. Therefore Y ⊆ Z or Z ⊆ Y . Thus Claim 2 is proven. Since C is a chain with X∈C yX χX = p and yX > 0 for X ∈ C \ {V }, the family C must coincide with {Ui | i = 1, . . . , m} (cf. (4.5)), and therefore, Ui ∈ dom ρ (i = 1, . . . , m). This completes the proof of Claim 1. Suppose that (B) is feasible, and let y ∗ be defined by (4.17) with reference to a maximal chain {Vj } of dom ρ containing {Ui }. Define another vector x∗ ∈ RV by x∗ (vj ) = ρ(Vj ) − ρ(Vj−1 )

(1 ≤ j ≤ n),

(4.18)

with V0 = ∅. The solutions x = x∗ and y = y ∗ are feasible in (A) and (B), respectively, and have the same objective value: ∗

p, x =

n 

p(vj )[ρ(Vj ) − ρ(Vj−1 )]

j=1

=

n−1 



j=1

X

[p(vj ) − p(vj+1 )]ρ(Vj ) + p(vn )ρ(V ) =

∗ yX ρ(X).

By of y ∗ for (B); hence LP∗ duality (Theorem 3.10 (1)), this shows the optimality ∗ y ρ(X) = min (B). On the other hand, y ρ(X) = ρˆ(p) by (4.6) and X X X X (4.17). Combining these with (4.16) shows (4.14).

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In the case of infeasible (B), we have min (B) = +∞ in (4.16), whereas ρˆ(p) = +∞ by Claim 1 and (4.8). Hence (4.14) follows. Proposition 4.6. B(ρ) is an integral polyhedron for ρ ∈ S[Z]. Proof. For each p, the optimal base (4.18) is an integer vector for ρ ∈ S[Z].

Note 4.7. A ring family (4.12) typically arises from a graph, and conversely, any ring family can be represented by a graph. For a directed graph G = (V, A) with vertex set V and arc set A, we call a subset X of V an ideal if (u, v) ∈ A and u ∈ X imply v ∈ X. That is, X is an ideal if and only if no arc leaves X. The set of ideals D = D(G) = {X ⊆ V | (u, v) ∈ A, u ∈ X ⇒ v ∈ X}

(4.19)

is a ring family with {∅, V } ⊆ D. Conversely, for a ring family D on a set V , we consider a directed graph G = (V, A) with A = A(D) = {(u, v) | u ∈ X ∈ D ⇒ v ∈ X}.

(4.20)

Denote by min D the minimum element of D and by max D the maximum element. Then D coincides with the family of ideals of G that contain min D and are contained in max D. In particular, D = D(G) if {∅, V } ⊆ D. The graph G given by (4.20) is transitive; i.e., (u, v) ∈ A, (v, w) ∈ A ⇒ (u, w) ∈ A. The constructions (4.19) and (4.20) establish a one-to-one correspondence between the set of ring families on V including {∅, V } and the set of transitive directed graphs with vertex set V . An acyclic graph44 G = (V, A) represents a partial order ( on V defined by [v ( u ⇔ v is reachable from u by a directed path], and the corresponding ring family D has the property that a maximal chain of D is of length |V |. In this particular case, (4.19) reads D = {X ⊆ V | v ( u, u ∈ X ⇒ v ∈ X}.

(4.21)

Thus, we have a one-to-one correspondence between the set of partial orders on V and the set of ring families on V including {∅, V } and having a maximal chain of length |V |. Note 4.8. The minimizers of a submodular set function ρ form a ring family. Let α denote the minimum of ρ. If ρ(X) = ρ(Y ) = α, we have 2α = ρ(X) + ρ(Y ) ≥ ρ(X ∪ Y ) + ρ(X ∩ Y ) ≥ 2α by submodularity (4.9). This implies ρ(X ∪ Y ) = ρ(X ∩ Y ) = α. Hence, X, Y ∈ arg min ρ =⇒ X ∪ Y, X ∩ Y ∈ arg min ρ. 44 A

directed graph is called acyclic if it does not contain directed cycles.

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Note 4.9. For a base x ∈ B(ρ), a subset X ⊆ V is said to be a tight set at x if x(X) = ρ(X). The family of tight sets at x, denoted by D(x) = {X ⊆ V | x(X) = ρ(X)},

(4.22)

is a ring family satisfying X, Y ∈ D(x) =⇒ X ∪ Y, X ∩ Y ∈ D(x).

(4.23)

This follows from Note 4.8 applied to ρ(X) − x(X). Note that {∅, V } ⊆ D(x). Note 4.10. LP (A) in the proof of Proposition 4.5 is the problem of maximizing the weight of a base x ∈ B(ρ) with respect to a given weight vector p. An optimum solution is given by (4.18) for an ordering of the elements of V satisfying p(v1 ) ≥ · · · ≥ p(vn ), where dom ρ is assumed to have a maximal chain of length n = |V |. Often we refer to this fact by saying that the greedy algorithm works for finding an optimal base. Note 4.11. A partial order on V is associated with each extreme base. Assume that D = dom ρ has a maximal chain of length n = |V | and denote by ( the partial order on V associated with D, as in (4.21). A linear order ≤ on V , or an ordering of elements of V , is said to be an extension of ( if [v ( u ⇒ v ≤ u]. A linear extension of ( generates an extreme base in (4.18). Conversely, any extreme base x is generated in this way, but there can be several linear orders that generate x. We define a partial order (x on V associated with x by v (x u ⇐⇒ v ≤ u for every linear order ≤ that generates x. The partially ordered set P(x) = (V, (x ) thus defined corresponds to the family of tight sets D(x) in the sense of Note 4.7. In particular, [x(X) = ρ(X)] ⇐⇒ [v (x u, u ∈ X ⇒ v ∈ X].

4.4

Polyhedral Description of M-Convex Sets

An M-convex set is hole free (Theorem 4.12 below), which allows us to identify an M-convex set with its convex hull. The convex hull of an M-convex set is called an M-convex polyhedron, which is indeed a polyhedron described by a submodular function (Proposition 4.13 below). Let us start with the hole-free property of an M-convex set. Theorem 4.12. B = B ∩ ZV for an M-convex set B ⊆ ZV . Proof. Obviously, B ⊆ B ∩ ZV . To show the reverse inclusion, take an arbitrary x ∈ B ∩ ZV , which can be represented as x=

m  i=1

λi xi ,

xi ∈ B, λi > 0 (1 ≤ i ≤ m),

m  i=1

λi = 1

(4.24)

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109

with distinct xi (1 ≤ i ≤ m). We may assume that there is a positive integer N such that N λi ∈ Z+ (1 ≤ i ≤ m). For a representation of the form (4.24), we define Φ=

m  i=1

λi ||xi − x||1 =

m  i=1

λi



|xi (v) − x(v)|,

v∈V

which is intended to measure the complexity of the representation. The representation (4.24) with m = 1 means x ∈ B (we are done). If m ≥ 2, there are two distinct indices j, k and u ∈ V such that xj (u) < x(u) < xk (u). (B-EXC[Z]) shows the existence of v ∈ supp− (xk − xj ) with xk = xk − χu + χv ∈ B and xj = xj + χu − χv ∈ B. A modification of the representation (4.24), according to the following: if λk ≥ λj : λj xj + λk xk ⇒ λj (xj + xk ) + (λk − λj )xk , if λj ≥ λk : λj xj + λk xk ⇒ λk (xj + xk ) + (λj − λk )xj , gives another representation of the form (4.24), for which Φ is smaller at least by 2 min(λj , λk ) (≥ 2/N ). The condition N λi ∈ Z+ is preserved in this modification. The process of modification ends with m = 1, showing x ∈ B. The convex hull of an M-convex set is a polyhedron—the base polyhedron defined by some submodular set function. Proposition 4.13. For an M-convex set B, define ρ : 2V → Z ∪ {+∞} by ρ(X) = sup{x(X) | x ∈ B}

(X ⊆ V ).

(4.25)

(1) ρ ∈ S[Z]. (2) B = B(ρ). Proof. (1) First we show the submodularity inequality (4.9) for X and Y with ρ(X ∪ Y ) and ρ(X ∩ Y ) both finite. Take y, z ∈ B with ρ(X ∪ Y ) = y(X ∪ Y ) and ρ(X ∩ Y ) = z(X ∩ Y ); we choose such (y, z) with ||y − z||1 minimum. Then we have y(v) = z(v) (v ∈ X ∩ Y ), since, otherwise, ∃ u ∈ (X ∩ Y ) ∩ supp+ (z − y) and, by (B-EXC+ [Z]), ∃ v ∈ supp− (z − y) such that y  = y + χu − χv ∈ B, for which we have y  (X ∪ Y ) ≥ y(X ∪ Y ) and ||y  − z||1 ≤ ||y − z||1 − 2, a contradiction to our choice of (y, z). Therefore, ρ(X ∪ Y ) + ρ(X ∩ Y ) = y(X ∪ Y ) + z(X ∩ Y ) = y(X ∪ Y ) + y(X ∩ Y ) = y(X) + y(Y ) ≤ ρ(X) + ρ(Y ), which shows (4.9). In the case of ρ(X ∪Y )+ ρ(X ∩Y ) = +∞, we consider a sequence of M-convex sets Bk = {x ∈ B | −k ≤ x(v) ≤ k (v ∈ V )} and the corresponding ρk ∈ S[Z] for k = 1, 2, . . . . The submodularity inequality (4.9) follows from ρ(X) + ρ(Y ) ≥ ρk (X) + ρk (Y ) ≥ ρk (X ∪ Y ) + ρk (X ∩ Y ) by letting k → +∞.

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(2) The inclusion B ⊆ B(ρ) is obvious. For the converse we may assume that ρ(X) is finite for all X ⊆ V (by the same argument as in the latter half of the proof of (1)). Let z ∈ RV be an extreme point of B(ρ). As we have seen in (4.18), there is an ordering of the elements of V , say, V = {v1 , . . . , vn }, such that z(Vj ) = ρ(Vj ) with Vj = {v1 , . . . , vj } for j = 1, . . . , n, where n = |V |. For each j = 1, . . . , n, there exists xj ∈ B with ρ(Vj ) = xj (Vj ). By repeated applications of (B-EXC+ [Z]), as in the proof of (1) above, we can show the existence of x ˆ ∈ B such that x ˆ(Vj ) = xj (Vj ) for j = 1, . . . , n. We then have z(Vj ) = ρ(Vj ) = xj (Vj ) = x ˆ(Vj ) for j = 1, . . . , n, which means z = x ˆ ∈ B. Since any extreme point of B(ρ) is contained in B, we must have B(ρ) ⊆ B. The converse of the above proposition is also true. Proposition 4.14. Let ρ ∈ S[Z] be an integer-valued submodular set function. (1) B = B(ρ) ∩ ZV is an M-convex set. (2) ρ(X) = sup{x(X) | x ∈ B(ρ)} (X ⊆ V ). Proof. (1) First, B is nonempty by Propositions 4.4 and 4.6. By Proposition 4.2 it suffices to show (B-EXC+ [Z]). Suppose, to the contrary, that (B-EXC+ [Z]) fails for some x, y ∈ B and some u ∈ supp+ (x − y). For each v ∈ supp− (x − y), we have y + χu − χv ∈ / B, which, together with the integrality of ρ, 4 implies the existence of a / Xv . For Z = v∈supp− (x−y) Xv , we have tight set Xv ∈ D(y) with u ∈ Xv and v ∈ y(Z) = ρ(Z) by (4.23), whereas x(Z) > y(Z) by u ∈ Z and Z ∩ supp− (x − y) = ∅. It then follows that x(Z) > ρ(Z), a contradiction to x ∈ B(ρ). (2) This follows from (4.14) with p = χX and (4.7). An alternative proof is as follows. Since ρ(X) ≥ sup{x(X) | x ∈ B(ρ)} is obvious, it suffices to establish an equality here in the case of sup < +∞. Let x ˆ ∈ B(ρ) attain the supremum. Then x) 5 with u4∈ Xuv and v ∈ / Xuv . for each u ∈ X and v ∈ V \ X there exists Xuv ∈ D(ˆ Since D(ˆ x) is a ring family (see (4.23)), we have X = u∈X v∈V \X Xuv ∈ D(ˆ x), which means x ˆ(X) = ρ(X). Propositions 4.13 and 4.14 together imply a one-to-one correspondence between the family M0 [Z] of M-convex sets and the family S[Z] of integer-valued submodular set functions. In this sense, the exchange property (B-EXC[Z]) and the submodularity (4.9) are equivalent. Theorem 4.15. A set B ⊆ ZV is M-convex if and only if B = B(ρ) ∩ ZV for an integer-valued submodular set function ρ ∈ S[Z]. More specifically, the mappings Φ : M0 [Z] → S[Z] and Ψ : S[Z] → M0 [Z] defined by Φ : B → ρ in (4.25),

Ψ : ρ → B = B(ρ) ∩ ZV

are inverse to each other, establishing a one-to-one correspondence between M0 [Z] and S[Z]. Proof. For B ∈ M0 [Z], we have Φ(B) ∈ S[Z] and Ψ ◦ Φ(B) = B ∩ ZV = B by

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Proposition 4.13 and Theorem 4.12, respectively. For ρ ∈ S[Z], we have Ψ(ρ) ∈ M0 [Z] and Φ ◦ Ψ(ρ) = ρ by Propositions 4.6 and 4.14.

4.5

Submodular Functions as Discrete Convex Functions

We prove two fundamental theorems connecting submodularity and convexity, which we have already seen in section 1.3.1. The Lov´asz extension ρˆ of a submodular set function ρ is a convex function, since, from the expression (4.14), ρˆ is the support function of the base polyhedron. The converse is also true. Theorem 4.16 (Lov´asz). A set function ρ : 2V → R ∪ {+∞} with ρ(∅) = 0 and ρ(V ) < +∞ is submodular if and only if its Lov´ asz extension, ρˆ : RV → R ∪ {+∞}, defined in (4.6) is convex; i.e., ρ is submodular ⇐⇒ ρˆ is convex. Proof. It suffices to prove ⇐ only. Since ρˆ is positively homogeneous, the convexity of ρˆ implies ρˆ(χX ) + ρˆ(χY ) ≥ ρˆ(χX + χY ). This shows the submodularity of ρ, since ρˆ(χX ) = ρ(X), ρˆ(χY ) = ρ(Y ), and ρˆ(χX + χY ) = ρ(X ∪ Y ) + ρ(X ∩ Y ) by (4.7) and (4.6). The connection of submodularity to convexity is reinforced by the following discrete separation theorem for a pair of submodular/supermodular set functions. Theorem 4.17 (Frank’s discrete separation theorem). Let ρ : 2V → R ∪ {+∞} and μ : 2V → R ∪ {−∞} be submodular and supermodular functions, respectively, with ρ(∅) = μ(∅) = 0, ρ(V ) < +∞, and μ(V ) > −∞ (namely, ρ, −μ ∈ S[R]). If ρ(X) ≥ μ(X)

(∀ X ⊆ V ),

(4.26)

there exists x∗ ∈ RV such that ρ(X) ≥ x∗ (X) ≥ μ(X)

(∀ X ⊆ V ).

(4.27)

Moreover, if ρ and μ are integer valued (namely, ρ, −μ ∈ S[Z]), the vector x∗ can be chosen to be integer valued (namely, x∗ ∈ ZV ). The combinatorial essence of the above theorem lies in the second half, claiming the existence of an integer vector for integer-valued functions, whereas the existence of a real vector x∗ alone can be proved on the basis of the separation theorem

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in convex analysis and the relationship between submodularity and convexity stated in Theorem 4.16 (see Note 4.20). The proof of Theorem 4.17 is based on Edmonds’s intersection theorem below, which is the most important duality theorem in the theory of submodular functions. For a submodular set function ρ ∈ S[R], we define a polyhedron P(ρ) = {x ∈ RV | x(X) ≤ ρ(X) (∀ X ⊆ V )},

(4.28)

called the submodular polyhedron associated with ρ. Note the relationship P(ρ) ∩ {x ∈ RV | x(V ) = ρ(V )} = B(ρ) to the base polyhedron B(ρ). Theorem 4.18 (Edmonds’s intersection theorem). Let ρ1 , ρ2 : 2V → R ∪ {+∞} be submodular set functions with ρ1 (∅) = ρ2 (∅) = 0, ρ1 (V ) < +∞, and ρ2 (V ) < +∞ (namely, ρ1 , ρ2 ∈ S[R]). Then max{x(V ) | x ∈ P(ρ1 ) ∩ P(ρ2 )} = min{ρ1 (X) + ρ2 (V \ X) | X ⊆ V }.

(4.29)

Moreover, if ρ1 and ρ2 are integer valued (namely, ρ1 , ρ2 ∈ S[Z]), the polyhedron P(ρ1 ) ∩ P(ρ2 ) is integral in the sense of P(ρ1 ) ∩ P(ρ2 ) = P(ρ1 ) ∩ P(ρ2 ) ∩ ZV

(4.30)

and there exists an integer-valued vector x∗ that attains the maximum on the lefthand side of (4.29). Proof. Denoting Di = (dom ρi ) \ {∅} (i = 1, 2), we consider a pair of LPs: (A)

Maximize subject to

(B)

Minimize subject to

p, x

χX , x ≤ ρ1 (X) (X ∈ D1 ), χX , x ≤ ρ2 (X) (X ∈ D2 ). {y1X ρ1 (X) | X ∈ D1 } + {y2X ρ2 (X) | X ∈ D2 } {y1X | v ∈ X ∈ D1 } + {y2X | v ∈ X ∈ D2 } = p(v) (v ∈ V ), y1X ≥ 0 (X ∈ D1 ), y2X ≥ 0 (X ∈ D2 ).

Here x ∈ RV and (yiX | X ∈ Di , i = 1, 2) are the variables of (A) and (B), respectively, and p ∈ RV is a parameter. Note that the equality constraints in (B) can be written as   y1X χX + y2X χX = p. (4.31) X∈D1

X∈D2

Problem (A) is feasible. Let p be such that (A) has an optimal solution. Then Problem (B) also has an optimal solution by LP duality (Theorem 3.10).

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There exists an optimal solution to (B) such that Ci = {X ∈ Di | yiX > 0} forms a chain for i = 1, 2. To prove this, take an optimal solution that maximizes   Γ= y1X |X|2 + y2X |X|2 . X∈D1

X∈D2

If yiY ≥ yiZ > 0 for some i ∈ {1, 2} and Y, Z ∈ Di , we have yiY χY + yiZ χZ = yiZ χY ∩Z + (yiY − yiZ )χY + yiZ χY ∪Z , yiY ρi (Y ) + yiZ ρi (Z) ≥ yiZ ρi (Y ∩ Z) + (yiY − yiZ )ρi (Y ) + yiZ ρi (Y ∪ Z), where the latter is due to the submodularity of ρi . This means that the modification  of (yiX ) to (yiX ) defined by ⎧ 0 (X = Z), ⎪ ⎪ ⎨ yiY − yiZ (X = Y ),  yiX = yiX + yiZ (X = Y ∪ Z, Y ∩ Z), ⎪ ⎪ ⎩ (otherwise) yiX would increase Γ by 2yiZ |Y \ Z| |Z \ Y | while maintaining the optimality. By the maximality of Γ we must have |Y \ Z| = 0 or |Z \ Y | = 0; i.e., Y ⊆ Z or Z ⊆ Y . Let Ai be the incidence matrix of Ci for i = 1, 2. Namely, Ai is a |Ci | × |V | matrix with rows indexed by Ci and columns by V ; for X ∈ Ci and v ∈ V , the 1 (X, v)-entry is equal to 1 if v ∈ X and to 0 otherwise. Define A = [ A A2 ], which is a totally unimodular matrix, as shown in Example 3.12. The vector y˜ = (yiX | X ∈ Ci , i = 1, 2) of nonzero entries of the optimal solution to (B) is determined as a solution to y˜ A = p; see (4.31). By the total unimodularity of A, y˜ can be chosen to be integral for an integral p. For p = 1, y˜ is a {0, 1}-vector, which implies C1 = {X} and C2 = {V \ X} for some X ⊆ V . Hence, the optimal value of Problem (B) for p = 1 is equal to the right-hand side of (4.29). On the other hand, the optimal value of Problem (A) for p = 1 is obviously equal to the left-hand side of (4.29). Then the identity (4.29) follows from LP duality (Theorem 3.10 (2)). It remains to show the integrality (4.30) for ρ1 , ρ2 ∈ S[Z]. Define a vector ρ˜ = (ρi (X) | X ∈ Ci , i = 1, 2). By the complementarity (Theorem 3.10 (3)), a feasible solution x to (A) is optimal if and only if it satisfies Ax = ρ˜. Such an x can be chosen to be integral by the integrality of ρ˜ and the total unimodularity of A. The above argument shows that Problem (A) has an integral optimal solution for any p ∈ ZV for which (A) has an optimal solution. This implies (4.30). As an immediate corollary of (4.30) we obtain B(ρ1 ) ∩ B(ρ2 ) = B(ρ1 ) ∩ B(ρ2 ) ∩ ZV

(4.32)

for ρ1 , ρ2 ∈ S[Z], the integrality of the intersection of integral base polyhedra. We are now in the position to prove Frank’s discrete separation theorem (Theorem 4.17). Consider Edmonds’s intersection theorem (Theorem 4.18) for ρ1 (X) = ρ(X),

ρ2 (X) = μ(V ) − μ(V \ X).

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It follows from (4.26) that the minimum on the right-hand side of (4.29) is equal to μ(V ). Hence, there exists x∗ ∈ P(ρ1 ) ∩ P(ρ2 ) such that x∗ (V ) = μ(V ). The condition x∗ ∈ P(ρ1 ) is equivalent to x∗ (X) ≤ ρ(X) (∀ X ⊆ V ), and x∗ ∈ P(ρ2 ) is equivalent to x∗ (V \ X) ≤ ρ2 (V \ X) (∀ X ⊆ V ), which is equivalent further to x∗ (X) ≥ μ(X) (∀ X ⊆ V ) by x∗ (V ) = μ(V ). Hence, ρ(X) ≥ x∗ (X) ≥ μ(X) for all X ⊆ V . For integer-valued ρ and μ, we can take an integral x∗ by the integrality assertion in Theorem 4.18. This completes the proof of Theorem 4.17. Note 4.19. Discreteness is twofold in Edmonds’s intersection theorem. First, the minimum on the right-hand side of (4.29) is taken over combinatorial objects, i.e., subsets of V , independently of whether the submodular functions are integer valued or not. Second, the maximum can be taken over discrete points in the case of integer-valued submodular functions. The former is sometimes referred to as the dual integrality and the latter as the primal integrality. Note 4.20. The first half of the discrete separation theorem (Theorem 4.17) is derived here from the separation theorem in convex analysis and the relationship between submodularity and convexity given in Theorem 4.16. Let ρˆ and μ ˆ be the Lov´ asz extensions of ρ and μ, respectively. We have ρˆ(p) ≥ μ ˆ (p) (∀ p ∈ RV+ ) by the assumption ρ ≥ μ as well as the definition (4.6) of the Lov´ asz extension. Define functions g : RV → R ∪ {+∞} and k : RV → R ∪ {−∞} by   μ ˆ(p) (p ∈ RV+ ), ρˆ(p) (p ∈ RV+ ), k(p) = g(p) = +∞ (otherwise), −∞ (otherwise). Then g is convex and k is concave by Theorem 4.16; these functions are polyhedral, and dom g ∩ dom k = ∅ by g(0) = k(0) = 0. The separation theorem in convex analysis (Theorem 3.5) applies to the pair of g and k, yielding β ∗ ∈ R and x∗ ∈ RV such that g(p) ≥ β ∗ + p, x∗ ≥ k(p)

(∀ p ∈ RV ).

This inequality for p = χX (X ⊆ V ) yields the inequality (4.27), where β ∗ = 0 follows from g(0) = ρ(∅) = 0 and k(0) = μ(∅) = 0.

4.6

M-Convex Sets as Discrete Convex Sets

We show a number of nice properties of M-convex sets that qualify them as wellbehaved discrete convex sets. We start with a discrete separation theorem for two M-convex sets. Theorem 4.21 (Discrete separation for M-convex sets). Let B1 and B2 (⊆ ZV ) be M-convex sets. If they are disjoint (B1 ∩B2 = ∅), there exists p∗ ∈ {0, 1}V ∪{0, −1}V such that inf{ p∗ , x | x ∈ B1 } − sup{ p∗ , x | x ∈ B2 } ≥ 1. (4.33)

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4.6. M-Convex Sets as Discrete Convex Sets

115

Proof. By Theorem 4.15, we have Bi = B(ρi ) ∩ ZV for some submodular functions ρi ∈ S[Z] (i = 1, 2). If ρ1 (V ) = ρ2 (V ), we can take p∗ = χV or −χV . Suppose ρ1 (V ) = ρ2 (V ). Theorem 4.17 applied to ρ(X) = ρ1 (X) and μ(X) = ρ2 (V )−ρ2 (V \ X) yields ρ1 (V ) = ρ2 (V ), B(ρ1 ) ∩ B(ρ2 ) ∩ ZV = ∅ ⇒ ∃ X ⊆ V : μ(X) − ρ(X) ≥ 1. Since B1 ∩ B2 = ∅ by assumption, there exists such an X. Noting ρ(X) = sup{ χX , x | x ∈ B1 },

μ(X) = inf{ χX , x | x ∈ B2 },

we see that p∗ = −χX (or p∗ = 1 − χX ∈ {0, 1}V ) is a valid choice for (4.33). The content of Theorem 4.21 consists of two claims. The first, explicit in the statement, is that the separating vector p∗ is so special that p∗ or −p∗ is a {0, 1}vector. The second, less conspicuous and more subtle, is that B1 ∩B2 = ∅ is implied by B1 ∩ B2 = ∅, since otherwise the inequality (4.33) is impossible. The implication B1 ∩ B2 = ∅ =⇒ B1 ∩ B2 = ∅

(4.34)

was named convexity in intersection in section 3.3. The following theorem shows another integrality property, stronger than (4.34), of the intersection of two M-convex polyhedra. Theorem 4.22. For M-convex sets B1 , B2 ⊆ ZV we have B1 ∩ B2 = B1 ∩ B2 . Proof. For the representation Bi = B(ρi ) ∩ ZV with ρi ∈ S[Z] (i = 1, 2), we have Bi = B(ρi ) (i = 1, 2). Then the claim follows from (4.32). We now turn to the Minkowski sum of M-convex sets. The claim (3) in the theorem below says that the Minkowski sum of M-convex sets is again an M-convex set. An important consequence of this is that the family of M-convex sets has the property of convexity in Minkowski sum considered in section 3.3. Theorem 4.23. (1) For submodular set functions ρ1 , ρ2 ∈ S[R], we have B(ρ1 ) + B(ρ2 ) = B(ρ1 + ρ2 ). (2) For integer-valued submodular set functions ρ1 , ρ2 ∈ S[Z], we have (B(ρ1 ) ∩ ZV ) + (B(ρ2 ) ∩ ZV ) = B(ρ1 + ρ2 ) ∩ ZV . (3) For M-convex sets B1 , B2 ⊆ ZV , B1 + B2 is an M-convex set and B1 + B2 = B1 + B2 ∩ ZV .

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Chapter 4. M-Convex Sets and Submodular Set Functions

Proof. (1) The proof of B(ρ1 ) + B(ρ2 ) ⊆ B(ρ1 + ρ2 ) is easy: A vector x ∈ B(ρ1 ) + B(ρ2 ) can be decomposed as x = x1 + x2 with xi ∈ B(ρi ) (i = 1, 2), which implies x(X) = x1 (X) + x2 (X) ≤ ρ1 (X) + ρ2 (X) (equality for X = V ). Conversely, for x ∈ B(ρ1 + ρ2 ), we have x(X) − ρ2 (X) ≤ ρ1 (X), and by the discrete separation theorem (Theorem 4.17), there exists y ∈ RV such that x(X) − ρ2 (X) ≤ y(X) ≤ ρ1 (X), with equality for X = V . For z = x − y, we have y ∈ B(ρ1 ) and z ∈ B(ρ2 ). Hence x = y + z ∈ B(ρ1 ) + B(ρ2 ). (2) This is because the vectors x, y, z above can be taken to be integral. (3) We can represent Bi = B(ρi ) ∩ ZV with ρi ∈ S[Z] (i = 1, 2). Then the left-hand side of (2) is B1 + B2 , whereas for the right-hand side of (2) we have B(ρ1 + ρ2 ) = B(ρ1 ) + B(ρ2 ) = B1 + B2 = B1 + B2 by (1) and Proposition 3.17 (4). Since ρ1 + ρ2 ∈ S[Z], B1 + B2 = B(ρ1 + ρ2 ) ∩ ZV is an M-convex set. Finally, we show the integral convexity of an M-convex set. Theorem 4.24. An M-convex set is integrally convex. Proof. Let B be an M-convex set and H = {x ∈ RV | x(V ) = r} be the hyperplane containing it. Theorem 4.22 applied to B1 = B and B2 = N (x) ∩ H with N (x) defined in (3.58) yields B ∩ N (x) ∩ H = B ∩ N (x) ∩ H. This implies B ∩ N (x) = B ∩ N (x) since B ∩ H = B, N (x) ∩ H = N (x) ∩ H, and B ∩ H = B. Then the integral convexity of B follows from (3.72).

Note 4.25. The intersection of two M-convex sets is not necessarily M-convex, though it is integrally convex (see Theorem 8.31). Such a set is referred to as an M2 -convex set . An example of an M2 -convex set that is not M-convex is given by S = {(0, 0, 0, 0), (1, 0, 0, −1), (0, 1, 0, −1), (0, 0, 1, −1), (1, 0, 1, −2)}, which is the intersection of two M-convex sets B1 = S ∪ {(0, 1, 1, −2)} and B2 = S ∪ {(1, 1, 0, −2)}. Note that (B-EXC[Z]) fails for S with x = (1, 0, 1, −2), y = (0, 1, 0, −1), and u = 1.

4.7

M -Convex Sets

In section 4.1 we introduced the concept of M -convex sets as the projection of Mconvex sets along an arbitrarily chosen coordinate axis. The concepts of M -convex sets and M-convex sets are essentially equivalent, since an M-convex set lies on a hyperplane {x ∈ RV | x(V ) = r} for some r ∈ Z (Proposition 4.1). All the results for M-convex sets can be translated for M -convex sets. Here we state some of these (nontrivial) translations. The definition of an M -convex set by projection may be stated more formally as follows. Let 0 denote a new element not in V and put V˜ = {0} ∪ V . A set

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4.7. M -Convex Sets

117

Figure 4.1. M -convex sets.

Q ⊆ ZV is an M -convex set if it can be represented as Q = {x ∈ ZV | (x0 , x) ∈ B}

(4.35)

for some M-convex set B ⊆ Z{0}∪V . It turns out that an M -convex set Q can be characterized by an exchange axiom: (B -EXC[Z]) For x, y ∈ Q and u ∈ supp+ (x − y), (i) x − χu ∈ Q and y + χu ∈ Q, or (ii) there exists v ∈ supp− (x − y) such that x − χu + χv ∈ Q and y + χu − χv ∈ Q. It is required in (B -EXC[Z]) that at least one of (i) and (ii) be satisfied, depending on a given triple (x, y, u). Examples of M -convex sets are given in Fig. 4.1. Whereas M -convex sets are conceptually equivalent to M-convex sets, the class of M -convex sets is strictly larger than that of M-convex sets. This follows from the implication (B-EXC[Z]) ⇒ (B -EXC[Z]), as well as from the example of an integer interval [a, b]Z that is not M-convex but M -convex. We denote by M0 [Z] the set of M -convex sets. The projection of a base polyhedron is known to coincide with what is called a generalized polymatroid (or g-polymatroid for short); see Theorem 3.58 of Fujishige [65]. Hence, an M -convex set is precisely the set of integer points of an integral g-polymatroid and the convex hull of an M -convex set (M -convex polyhedron) is represented as Q(ρ, μ) = {x ∈ RV | μ(X) ≤ x(X) ≤ ρ(X) (∀ X ⊆ V )}

(4.36)

for the pair (ρ, μ) of integer-valued submodular function ρ ∈ S[Z] and supermodular function μ (i.e., −μ ∈ S[Z]) such that ρ(X) − ρ(X \ Y ) ≥ μ(Y ) − μ(Y \ X)

(X, Y ⊆ V ).

(4.37)

A set Q ⊆ ZV satisfies (B -EXC[Z]) if and only if it can be represented as Q = Q(ρ, μ) ∩ ZV in this way. The intersection of two M -convex sets is called an M2 -convex set . An M2 convex set is a projection of an M2 -convex set, and the class of M2 -convex sets is strictly larger than that of M2 -convex sets.

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118

4.8

Chapter 4. M-Convex Sets and Submodular Set Functions

M-Convex Polyhedra

M-convex polyhedra are defined in section 4.4 as the convex hull of M-convex sets, and as such, they are necessarily integral polyhedra. The concept of M-convexity, however, can also be defined for general (nonintegral) polyhedra. A nonempty polyhedron B ⊆ RV is defined to be an M-convex polyhedron if it satisfies the following: (B-EXC[R]) For x, y ∈ B and u ∈ supp+ (x − y), there exist v ∈ supp− (x−y) and a positive number α0 ∈ R++ such that x−α(χu −χv ) ∈ B and y + α(χu − χv ) ∈ B for all α ∈ [0, α0 ]R . The following weaker exchange axiom: (B-EXC+ [R]) For x, y ∈ B and u ∈ supp+ (x − y), there exist v ∈ supp− (x−y) and a positive number α0 ∈ R++ such that y+α(χu −χv ) ∈ B for all α ∈ [0, α0 ]R , is in fact equivalent to (B-EXC[R]). That is, (B-EXC[R]) ⇐⇒ (B-EXC+ [R])

(4.38)

for a nonempty polyhedron B ⊆ RV (cf. Proposition 4.2). The one-to-one correspondence of M-convex sets with submodular set functions (Theorem 4.15) is generalized as follows: B is an M-convex polyhedron ⇐⇒ B = B(ρ) for ρ ∈ S[R], (4.39) B is an integral M-convex polyhedron ⇐⇒ B = B(ρ) for ρ ∈ S[Z], (4.40) where an integral M-convex polyhedron means an M-convex polyhedron B such that B = B ∩ ZV . For an integral M-convex polyhedron B and integer points x, y ∈ B ∩ZV , we can take α0 = 1 in (B-EXC[R]) by (4.40). An integral polyhedron B is M-convex if and only if B ∩ ZV is an M-convex set. We denote by M0 [R] the set of M-convex polyhedra and by M0 [Z|R] the set of integral M-convex polyhedra. The projection of an M-convex polyhedron along a coordinate axis is called an M -convex polyhedron, for which we have Q ⊆ RV is an M -convex polyhedron ⇐⇒ Q satisfies (B -EXC[R]) ⇐⇒ Q = Q(ρ, μ) for (ρ, μ) with ρ, −μ ∈ S[R] and (4.37),

(4.41)

where Q(ρ, μ) is defined in (4.36), and the following exchange axiom: (B -EXC[R]) For x, y ∈ Q and u ∈ supp+ (x − y), there exist v ∈ supp− (x − y) ∪ {0} and a positive number α0 ∈ R++ such that x − α(χu − χv ) ∈ Q and y + α(χu − χv ) ∈ Q for all α ∈ [0, α0 ]R . We denote by M0 [R] and M0 [Z|R] the sets of M -convex polyhedra and integral M -convex polyhedra, respectively.

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4.8. M-Convex Polyhedra

119

An M-convex cone means a cone that is an M-convex polyhedron. It is characterized as a convex cone spanned by vectors of the form χu − χv (u, v ∈ V ), to be proved in Note 8.9. That is, ⎧ ) ⎫ ) ⎨  ⎬ ) B is an M-convex cone ⇐⇒ B = cuv (χu − χv ))) cuv ≥ 0 ((u, v) ∈ A) ⎩ ⎭ ) (u,v)∈A for some A ⊆ V × V ,

(4.42)

where we may assume that A is transitive; i.e., (u, v) ∈ A, (v, w) ∈ A ⇒ (u, w) ∈ A. See Theorem 3.26 of Fujishige [65] for the extreme rays of an M-convex cone. An M-convex polyhedron is characterized as a polyhedron such that the tangent cone at each point is an M-convex cone (by (a) ⇔ (b) in Theorem 6.63). Combining this with (4.42) yields a characterization of an M-convex polyhedron in terms of the direction of edges: B is an M-convex polyhedron ⇐⇒ each edge of B is parallel to χu − χv for some u, v ∈ V .

(4.43)

Similarly, Q is an M -convex polyhedron ⇐⇒ each edge of Q is parallel to χu − χv for some u, v ∈ V ∪ {0}, (4.44) where χ0 = 0. It is noted again that M-convex polyhedra and M -convex polyhedra are synonyms of base polyhedra and g-polymatroids.

Bibliographical Notes As remarked already, this chapter is a reorganization of known results in the theory of submodular functions; see Fujishige [65] and Schrijver [183]. The proof of Theorem 4.12 (hole-free property) is taken from Murota [141]. The equivalence of exchangeability and submodularity (Theorem 4.15) is well known, but neither precise statement nor proof can be found in the literature; Theorem 4.15 and the proof are taken from [141]. The name “Lov´asz extension” was introduced by Fujishige [63], [65]. Theorem 4.16 (submodularity vs. convexity) is by Lov´ asz [123] and Theorem 4.17 (discrete separation theorem) by Frank [55]. The intersection theorem (Theorem 4.18) and the related statements (Theorem 4.22 (convexity in intersection) and Theorem 4.23 (convexity in Minkowski sum)) are due to Edmonds [44]. Theorem 4.21 (separation of M-convex sets) is observed in Murota [140]. Integral convexity of M-convex sets (Theorem 4.24) is due to Murota–Shioura [153]. The example in Note 4.25 is an adaptation of Example 3.7 of [153]. The terminology of M -convex sets as well as that of the exchange axiom (B -EXC[Z]) is introduced by Murota–Shioura [151]. The concept of g-polymatroids is due to Frank [57] (see also Frank–Tardos [58]), whereas the characterization as the projection of base polyhedra is by Fujishige [64]. Proofs of (4.38) and (4.39) can be

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Chapter 4. M-Convex Sets and Submodular Set Functions

found in Murota–Shioura [152]. The characterization (4.43) of base polyhedra by edges seems to have appeared first in Tomizawa [200] (without proof); a proof can be found in Fujishige–Yang [69] and an alternative proof is given in Note 8.9. Variants of (4.43) yield other classes of polyhedra with combinatorial structures; see Danilov–Koshevoy [32], Fujishige–Makino–Takabatake–Kashiwabara [67], and Kashiwabara–Takabatake [109]. A relaxation (weakening) of the exchange axiom gives rise to the concept of the jump systems of Bouchet–Cunningham [18]; see also Lov´ asz [124].

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Chapter 5

L-Convex Sets and Distance Functions

L-convex sets form another class of well-behaved discrete convex sets. They are defined in terms of an abstract axiom and correspond one-to-one to integer-valued distance functions satisfying the triangle inequality. L-convex sets (or their convex hull) are, in fact, a familiar object in the theory of network flows, though the terminology of L-convexity is not used there. Emphasis here is placed on a systematic presentation of various properties of L-convex sets from the viewpoint of discrete convex analysis.

5.1

Definition

A nonempty set of integer points D ⊆ ZV is defined to be an L-convex set if it satisfies the following two conditions: (SBS[Z]) (TRS[Z])

p, q ∈ D =⇒ p ∨ q, p ∧ q ∈ D, p ∈ D =⇒ p ± 1 ∈ D.

We denote by L0 [Z] the set of L-convex sets. L-convexity thus defined for a set D ⊆ ZV is equivalent to the L-convexity of the indicator function δD : ZV → {0, +∞} of D, defined in (3.51). Namely, D is an L-convex set, satisfying (SBS[Z]) and (TRS[Z]), if and only if δD is an L-convex function, satisfying (SBF[Z]) and (TRF[Z]) introduced in section 1.4.1. Since an L-convex set is homogeneous in the direction of 1 by (TRS[Z]), we may consider the restriction of an L-convex set to the coordinate plane defined by p(v0 ) = 0 for an arbitrary v0 ∈ V . A set derived from an L-convex set by such a restriction (intersection with a coordinate plane) is called an L -convex set . Whereas L -convex sets are conceptually equivalent to L-convex sets, the class of L -convex sets is strictly larger than that of L-convex sets. The simplest example of an L -convex set that is not L-convex is an integer interval [a, b]Z . We focus on L-convex sets in the development of the theory and deal with L -convex sets in section 5.5. 121

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122

5.2

Chapter 5. L-Convex Sets and Distance Functions

Distance Functions and Associated Polyhedra

We introduce here some fundamental facts about distance functions and the associated polyhedra, which turn out to be the convex hull of L-convex sets. By a distance function we mean a function γ : V × V → R ∪ {+∞} such that γ(v, v) = 0 (∀ v ∈ V ), where γ may take negative values and is not necessarily symmetric (i.e., γ(u, v) = γ(v, u) in general). With a distance function γ we can associate a directed graph Gγ = (V, Aγ ) with vertex set V and arc set Aγ = {(u, v) | γ(u, v) < +∞},

(5.1)

where γ(u, v) represents the length of arc (u, v). We denote by γ(u, v) the shortest length of a path from u to v in Gγ . The function γ is well defined if there exists no negative cycle in Gγ , where a negative cycle means a directed cycle of negative length. The triangle inequality γ(v1 , v2 ) + γ(v2 , v3 ) ≥ γ(v1 , v3 )

(∀ v1 , v2 , v3 ∈ V )

(5.2)

is a natural property for a distance function γ. We denote by T [R] the set of distance functions satisfying the triangle inequality and by T [Z] the set of integervalued such functions. We have γ ∈ T [R] for any distance function γ such that Gγ contains no negative cycle and γ = γ for γ ∈ T [R]. For a distance function γ, a vector p ∈ RV is said to be an admissible potential or a feasible potential if it satisfies the system of inequalities p(v) − p(u) ≤ γ(u, v)

(∀ u, v ∈ V, u = v).

(5.3)

The set of admissible potentials is denoted by D(γ) = {p ∈ RV | p(v) − p(u) ≤ γ(u, v) (∀ u, v ∈ V, u = v)}.

(5.4)

Note that the triangle inequality (5.2) is not assumed here. The following are fundamental facts well known in network flow theory. Proposition 5.1. Let γ be a distance function. (1) D(γ) = ∅ ⇐⇒ no negative cycle exists in graph Gγ . (2) If D(γ) = ∅, we have γ(u, v) = sup{p(v) − p(u) | p ∈ D(γ)}

(u, v ∈ V )

(5.5)

(u, v ∈ V ).

(5.6)

and D(γ) = D(γ). (3) For γ ∈ T [R], D(γ) is nonempty and γ(u, v) = sup{p(v) − p(u) | p ∈ D(γ)}

(4) D(γ) is an integral polyhedron for an integer-valued γ.

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5.3. Polyhedral Description of L-Convex Sets

123

Proof. For x ∈ RV we consider a pair of linear programs (LPs): (P)

Minimize

 

λuv γ(u, v)

(u,v)∈Aγ

subject to

λuv (χv − χu ) = x,

(u,v)∈Aγ

λuv ≥ 0 ((u, v) ∈ Aγ ).

(D)

Maximize subject to

p, x

p(v) − p(u) ≤ γ(u, v) ((u, v) ∈ Aγ ).

Here λ = (λuv | (u, v) ∈ Aγ ) and p are the variables of (P) and (D), respectively. The coefficient matrix is totally unimodular, being the negative of the incidence matrix of graph Gγ (see Example 3.11). The set of feasible solutions to (D) coincides with D(γ). (1) If (D) is feasible, the sum of the inequalities (5.3) for arcs in a directed cycle shows the nonnegativity of the cycle. Conversely, if no negative cycle exists, we can define p(v) to be the shortest path length from a fixed starting vertex to v to obtain a feasible solution p to (D) (with an obvious modification for vertices v not reachable from the starting vertex). (2) In the particular case of x = χv0 −χu0 for distinct u0 , v0 ∈ V , the objective function of (D) is equal to p, x = p(v0 ) − p(u0 ), and hence the optimal value of (D) equals the right-hand side of (5.5) for (u, v) = (u0 , v0 ). By Theorem 3.13, the optimal solution λ to (P) can be chosen to be an integer vector, which is in fact a {0, 1}-vector. Such an optimal solution to (P) represents a shortest path from u0 to v0 , and therefore, the optimal value of (P) is equal to γ(u0 , v0 ). By the feasibility of (D), LP duality (Theorem 3.10 (2)) applies to show (5.5). By γ ≥ γ we have D(γ) ⊇ D(γ). For p ∈ D(γ), adding the inequalities (5.3) for arcs in the shortest path from u0 to v0 yields p(v0 ) − p(u0 ) ≤ γ(u0 , v0 ), which shows D(γ) ⊆ D(γ). (3) The condition γ ∈ T [R] implies the nonexistence of negative cycles and γ = γ in (5.5). (4) By Theorem 3.13, the integrality of D(γ) follows from the total unimodularity of the coefficient matrix.

5.3

Polyhedral Description of L-Convex Sets

An L-convex set is hole free (Theorem 5.2 below), which allows us to identify an L-convex set with its convex hull. The convex hull of an L-convex set is called an L-convex polyhedron, which is indeed a polyhedron described by a distance function (Proposition 5.3 below). Let us start with the hole-free property of an L-convex set. Theorem 5.2. D = D ∩ ZV for an L-convex set D ⊆ ZV .

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Chapter 5. L-Convex Sets and Distance Functions

Proof. Obviously, D ⊆ D ∩ ZV . To show the reverse inclusion, take an arbitrary p ∈ D ∩ ZV , which can be represented as p=

m 

λi pi ,

pi ∈ D, λi > 0 (1 ≤ i ≤ m),

i=1

m 

λi = 1,

(5.7)

i=1

with distinct pi (1 ≤ i ≤ m). The representation (5.7) with m = 1 means p ∈ D (we are done). When m ≥ 2, repeated modifications of (5.7) as if λk ≥ λj : λj pj + λk pk ⇒ λj [(pj ∨ pk ) + (pj ∧ pk )] + (λk − λj )pk result in another representation of the form (5.7), with p1 ≤ p2 ≤ · · · ≤ pm . Then we have p1 ≤ p ≤ pm , in particular, and another kind of modification is applicable to (5.7): if λm ≥ λ1 : λ1 p1 + λm pm ⇒ λ1 (p1 + pm ) + (λm − λ1 )pm , if λ1 ≥ λm : λ1 p1 + λm pm ⇒ λm (p1 + pm ) + (λ1 − λm )p1 , where p1 = p1 + 1 and pm = pm − 1. Using these modifications we eventually arrive at (5.7) such that p − 1 ≤ p1 ≤ p ≤ pm ≤ p + 1. Then p1 = p − χX and pm = p + χX for some X ⊆ V , and hence p = (p1 + 1) ∧ pm ∈ D by (SBS[Z]) and (TRS[Z]). The convex hull of an L-convex set is a polyhedron described by some distance function. Proposition 5.3. For nonempty D ⊆ ZV , define γ : V × V → Z ∪ {+∞} by γ(u, v) = sup{p(v) − p(u) | p ∈ D}

(u, v ∈ V ).

(5.8)

(1) γ satisfies the triangle inequality (5.2); i.e., γ ∈ T [Z]. (2) D = D(γ) if D is an L-convex set. Proof. (1) γ(v1 , v2 ) + γ(v2 , v3 ) = supp∈D (p(v2 ) − p(v1 )) + supp∈D (p(v3 ) − p(v2 )) ≥ supp∈D (p(v3 ) − p(v1 )) = γ(v1 , v3 ). (2) Obviously, D ⊆ D(γ). By the integrality of D(γ) shown in Proposition 5.1 (4), the converse (⊇) is also true if any q ∈ D(γ) ∩ ZV belongs to D. For distinct u and v, we have γ(u, v) ≥ q(v)− q(u), and, by the definition of γ and (TRS[Z]), 6 there exists puv ∈ D such that puv (u) = q(u) and puv (v) ≥ q(v). For pu = v =u puv , we have pu (u) 7 = q(u) and pu (v) ≥ q(v) (∀ v ∈ V ), and also pu ∈ D by (SBS[Z]). Hence, for pˆ = u∈V pu , we have pˆ = q and also pˆ ∈ D by (SBS[Z]). A sort of converse of the above proposition is true. Note that the triangle inequality (5.2) is not assumed in the proposition below. Proposition 5.4. For an integer-valued distance function γ, D = D(γ) ∩ ZV is an L-convex set provided that D(γ) is nonempty.

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5.4. L-Convex Sets as Discrete Convex Sets

125

Proof. (TRS[Z]) is obvious. For (SBS[Z]) we can easily show that p(v) − p(u) ≤ γ(u, v) and q(v) − q(u) ≤ γ(u, v) imply (p ∨ q)(v) − (p ∨ q)(u) ≤ γ(u, v) and (p ∧ q)(v) − (p ∧ q)(u) ≤ γ(u, v). Propositions 5.3 and 5.4 together imply a one-to-one correspondence between the family L0 [Z] of L-convex sets and the family T [Z] of integer-valued distance functions with the triangle inequality. Theorem 5.5. A set D ⊆ ZV is L-convex if and only if D = D(γ) ∩ ZV for an integer-valued distance function γ ∈ T [Z] satisfying the triangle inequality. More specifically, the mappings Φ : L0 [Z] → T [Z] and Ψ : T [Z] → L0 [Z] defined by Φ : D → γ in (5.8),

Ψ : γ → D = D(γ) ∩ ZV

are inverse to each other, establishing a one-to-one correspondence between L0 [Z] and T [Z]. Proof. For D ∈ L0 [Z] we have Φ(D) ∈ T [Z] and Ψ ◦ Φ(D) = D ∩ ZV = D by Proposition 5.3 and Theorem 5.2. For γ ∈ T [Z] we have Ψ(γ) ∈ L0 [Z] and Φ ◦ Ψ(γ) = γ by Propositions 5.4 and 5.1.

Note 5.6. An M-convex polyhedron is described by a submodular set function (Theorem 4.15), and the correspondence is one-to-one: M-convex polyhedra ←→ submodular set functions. An L-convex polyhedron is described by a distance function with the triangle inequality (Theorem 5.5), which gives another one-to-one correspondence: L-convex polyhedra ←→ distance functions with the triangle inequality. These two one-to-one correspondences will be unified into a single conjugacy relationship between M-convex functions and L-convex functions in Chapter 8.

5.4

L-Convex Sets as Discrete Convex Sets

We show a number of nice properties of L-convex sets that qualify them as wellbehaved discrete convex sets. First we consider the intersection of two L-convex sets. Recall from (5.1) and (5.4) the definitions of a graph Gγ and a polyhedron D(γ) associated with a distance function γ. Theorem 5.7. Let D1 , D2 ⊆ ZV be L-convex sets. (1) D1 ∩ D2 = D1 ∩ D2 . On representing Di = D(γi ) ∩ ZV with γi ∈ T [Z] (i = 1, 2) and defining γ12 (u, v) = min(γ1 (u, v), γ2 (u, v)), we have the following. (2) D1 ∩ D2 = D(γ12 ) ∩ ZV .

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126

Chapter 5. L-Convex Sets and Distance Functions  ∅ ⇐⇒ no negative cycle exists in graph Gγ12 . (3) D1 ∩ D2 = (4) D1 ∩ D2 is an L-convex set if it is nonempty.

Proof. (1), (2) It follows from D(γ1 ) ∩ D(γ2 ) = D(γ12 ) that D1 ∩ D2 = (D(γ1 ) ∩ ZV ) ∩ (D(γ2 ) ∩ ZV ) = (D(γ1 ) ∩ D(γ2 )) ∩ ZV = D(γ12 ) ∩ ZV . Since D(γ12 ) is an integral polyhedron, we obtain D1 ∩ D2 = D(γ12 ) = D(γ1 ) ∩ D(γ2 ) = D1 ∩ D2 . (3) This is by (2) and Proposition 5.1 (1). (4) This follows from (2) and Proposition 5.4. The first claim (1) in the above theorem shows D1 ∩ D2 = ∅ =⇒ D1 ∩ D2 = ∅,

(5.9)

the property called convexity in intersection in section 3.3. Convexity in Minkowski sum is also shared by L-convex sets. Theorem 5.8. For L-convex sets D1 , D2 ⊆ ZV , we have D1 +D2 = D1 + D2 ∩ZV . Proof. F = L0 [Z] meets the condition (3.53) in Proposition 3.16 and has the property (a) there by (5.9). The following discrete separation theorem holds for two L-convex sets. Theorem 5.9 (Discrete separation for L-convex sets). Let D1 and D2 (⊆ ZV ) be L-convex sets. If they are disjoint (D1 ∩ D2 = ∅), there exists x∗ ∈ {−1, 0, 1}V such that (5.10) inf{ p, x∗ | p ∈ D1 } − sup{ p, x∗ | p ∈ D2 } ≥ 1. Proof. We use the notation in Theorem 5.7; in particular, Di = D(γi ) ∩ ZV with γi ∈ T [Z] (i = 1, 2). By Theorem 5.7 (3), there exists in the graph Gγ12 a negative cycle with respect to arc length γ12 = min(γ1 , γ2 ). Let v0 , v1 , v2 , . . . , vk−1 be the sequence of vertices in a negative cycle with the minimum number of vertices, where k ≥ 2. Since γ1 and γ2 satisfy the triangle inequality, k is even, and we may assume γ1 (v2i , v2i+1 ) ≤ γ2 (v2i , v2i+1 ) and γ1 (v2i+1 , v2i+2 ) ≥ γ2 (v2i+1 , v2i+2 ) for 0 ≤ i ≤ k/2 − 1, where vk = v0 . Define x∗ ∈ {−1, 0, 1}V by x∗ (v2i ) = 1, x∗ (v2i+1 ) = −1 (0 ≤ i ≤ k/2 − 1), and x∗ (v) = 0 for other v. It follows from LP duality (Theorem 3.10 (2)) and the minimality of k that 

k/2−1 ∗

inf p, x = −

p∈D1



k/2−1

γ1 (v2i , v2i+1 ),



sup p, x =

p∈D2

i=0

γ2 (v2i+1 , v2i+2 ),

i=0

and, therefore, 

k/2−1 ∗



sup p, x − inf p, x =

p∈D2

p∈D1

i=0



k/2−1

γ1 (v2i , v2i+1 ) +

i=0

γ2 (v2i+1 , v2i+2 )

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5.4. L-Convex Sets as Discrete Convex Sets

=

k−1 

127

γ12 (vi , vi+1 ) ≤ −1.

i=0

This shows (5.10). The content of Theorem 5.9 consists of two claims. The first, explicit in the statement, is that the separating vector x∗ is so special that it is a {0, ±1}-vector. The second, less conspicuous and more subtle, is the implication (5.9), convexity in intersection, since otherwise the inequality (5.10) is impossible. Finally, we show the integral convexity of an L-convex set by deriving an expression of the convex hull of an L-convex set. For a vector p ∈ RV , let α1 > α2 > · · · > αm be the distinct values of the nonzero components of vector a = p − p, and define Ui = Ui (p) = {v ∈ V | a(v) ≥ αi }

(i = 1, . . . , m),

where m ≥ 0. Then we have p=

m−1 

(αi − αi+1 )( p + χUi ) + αm ( p + χUm )

(5.11)

i=0

with α0 = 1 and U0 = ∅. This is a representation of p as a convex combination of p + χUi (i = 0, 1, . . . , m), since αi − αi+1 > 0 (i = 0, 1, . . . , m − 1), αm > 0, and m−1 i=0 (αi − αi+1 ) + αm = 1. Note that these points p + χUi (i = 0, 1, . . . , m) belong to the integral neighborhood N (p) of p defined by (3.58). The convex hull of an L-convex set can be characterized with reference to the expression (5.11). Theorem 5.10. For an L-convex set D ⊆ ZV , we have D = {p ∈ RV | p + χUi (p) ∈ D (i = 0, 1, . . . , m)}.

(5.12)

Hence, an L-convex set is integrally convex. Proof. The expression (5.11) shows the inclusion ⊇ in (5.12). To show the converse, take p ∈ D and put p0 = p, a = p − p, and qi = p + χUi (p) (i = 0, 1, . . . , m). In the representation D = D(γ) with an integer-valued distance function γ (by Theorem 5.5), we have p(v) − p(u) = [p0 (v) − p0 (u)] + [a(v) − a(u)] ≤ γ(u, v)

(∀ u, v ∈ V ).

Since |a(v) − a(u)| < 1, p0 (v) − p0 (u) ∈ Z, and γ(u, v) ∈ Z, we have p0 (v) − p0 (u) ≤ γ(u, v) for any u, v, and furthermore p0 (v) − p0 (u) + 1 ≤ γ(u, v) if a(v) > a(u). Hence follows qi (v) − qi (u) = [p0 (v) − p0 (u)] + [χUi (v) − χUi (u)] ≤ γ(u, v)

(∀ u, v ∈ V ),

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Chapter 5. L-Convex Sets and Distance Functions

which shows qi ∈ D(γ), and therefore, qi ∈ D(γ) ∩ ZV = D. Hence, we have ⊆ in (5.12). Since qi ∈ N (p) for i = 0, 1, . . . , m, we have p ∈ D ∩ N (p); see (3.71). Thus D is integrally convex.

Note 5.11. The Minkowski sum of two L-convex sets, to be called an L2 -convex set , is not necessarily L-convex, though it is integrally convex (see Theorem 8.42). An example of an L2 -convex set that is not L-convex is given by S = {(0, 0, 0, 0), (0, 1, 1, 0), (1, 1, 0, 0), (1, 2, 1, 0)} + {α1 | α ∈ Z}, which is the Minkowski sum of two L-convex sets D1 = {(0, 0, 0, 0), (1, 1, 0, 0)} + {α1 | α ∈ Z} and D2 = {(0, 0, 0, 0), (0, 1, 1, 0)} + {α1 | α ∈ Z}; note that (SBS[Z]) fails for (0, 1, 1, 0) and (1, 1, 0, 0).

5.5

L -Convex Sets

In section 5.1 we introduced the concept of L -convex sets as the restriction of Lconvex sets to an arbitrarily chosen coordinate plane. The concepts of L -convex sets and L-convex sets are essentially equivalent, since an L-convex set is homogeneous in the direction of 1 by (TRS[Z]). All the results for L-convex sets can be translated for L -convex sets. Here we state some additional results. The definition of an L -convex set by restriction may be stated more formally as follows. Let 0 denote a new element not in V and put V˜ = {0} ∪ V . A set P ⊆ ZV is an L -convex set if it can be represented as P = {p ∈ ZV | (0, p) ∈ D}

(5.13)

˜

for some L-convex set D ⊆ ZV . It turns out that an L -convex set P can be characterized by the property (SBS [Z])

p, q ∈ P =⇒ (p − α1) ∨ q, p ∧ (q + α1) ∈ P

(∀ α ∈ Z+ )

(see Note 5.12 for the proof). This condition for α = 0 agrees with (SBS[Z]). Examples of L -convex sets are given in Fig. 5.1. Whereas L -convex sets are conceptually equivalent to L-convex sets, the class  of L -convex sets is strictly larger than that of L-convex sets. This follows from the implication [(SBS[Z]) and (TRS[Z])] ⇒ (SBS [Z]), as well as from the example of an integer interval [a, b]Z that is not L-convex but L -convex. We denote by L0 [Z] the set of L -convex sets. For a set P ⊆ ZV , (SBS [Z]) above is equivalent to either of the following conditions: p, q ∈ P, supp+ (p − q) = ∅ =⇒ p − χX , q + χX ∈ P with X = arg max{p(v) − q(v)}, v∈V     p+q p+q p, q ∈ P =⇒ , ∈P 2 2

(5.14) (5.15)

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5.5. L -Convex Sets

129

Figure 5.1. L -convex sets.

p+q

q q

p+q

2

p+q p

2

p+q

2

2

p+q 2

q

p+q p

p

2

Figure 5.2. Discrete midpoint convexity.

(see Note 5.12 for the proof). The property (5.15) is called discrete midpoint convexity (see Fig. 5.2), where z and z denote, respectively, the integer vectors obtained from z ∈ RV by componentwise round-up and round-down to the nearest integer. Thus, L -convex sets can be characterized by one of the three equivalent conditions (SBS [Z]), (5.14), and (5.15). The convex hull of an L -convex set (called an L -convex polyhedron) can be represented as P(γ, γˆ, γˇ ) = {p ∈ RV | γˇ (v) ≤ p(v) ≤ γˆ (v) (∀ v ∈ V ), p(v) − p(u) ≤ γ(u, v) (∀ u, v ∈ V, u = v)},

(5.16)

with an integer-valued distance function γ : V × V → Z ∪ {+∞} and integer-valued functions γˆ : V → Z∪{+∞} and γˇ : V → Z∪{−∞}. We may impose an additional condition on (γ, γˆ , γˇ ): the distance function γ˜ on V˜ = V ∪ {0} defined by ⎧ ⎨ γ(u, v) (u, v ∈ V ), γ˜ (u, v) = γˆ (v) (u = 0, v ∈ V ), (5.17) ⎩ −ˇ γ (u) (v = 0, u ∈ V ) (as well as by γ˜ (v, v) = 0 (∀ v ∈ V˜ )) satisfies the triangle inequality. A set P ⊆ ZV satisfies (SBS [Z]) if and only if it can be represented as P = P(γ, γˆ , γˇ ) ∩ ZV . The Minkowski sum of two L -convex sets is called an L2 -convex set . An L2 convex set is a restriction of an L2 -convex set, and the class of L2 -convex sets is strictly larger than that of L2 -convex sets.

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Chapter 5. L-Convex Sets and Distance Functions

Note 5.12. We prove the equivalence among L -convexity (as induced from Lconvexity by restriction), (SBS [Z]), (5.14), and (5.15) for P ⊆ ZV . [L -convexity ⇔ (SBS [Z])]: By (5.13) and (TRS[Z]) we have (p0 , p) ∈ D ⇐⇒ p − p0 1 ∈ P. (SBS[Z]) for D is equivalent to the following condition for P : p − p0 1, q − q0 1 ∈ P =⇒ (p ∨ q) − (p0 ∨ q0 )1, (p ∧ q) − (p0 ∧ q0 )1 ∈ P. Assuming α = q0 − p0 ≥ 0, put p = p − p0 1 and q  = q − q0 1. Then (p ∨ q) − (p0 ∨ q0 )1 = (p − α1) ∨ q  and (p ∧ q) − (p0 ∧ q0 )1 = p ∧ (q  + α1). Hence, the above condition is equivalent to (SBS [Z]). [(SBS [Z])⇒(5.14)]: For α = maxv∈V {p(v) − q(v)} − 1, we have α ≥ 0, (p −  [Z]) implies (5.14). α1) ∨ q = q + χX , and p ∧ (q + α1)

p+q= p − χX. Then p+q(SBS  [(5.14)⇒(5.15)]: Put p = 2 and q = 2 , and define p , q  ∈ ZV by p (v) =



p (v) q  (v)

(p(v) ≥ q(v)), (p(v) ≤ q(v)),

q  (v) =



q  (v) p (v)

(p(v) ≥ q(v)), (p(v) ≤ q(v)).

Note that |p (v)−q  (v)| ≤ 1 (∀ v ∈ V ), supp+ (p −q  ) ⊆ supp+ (p−q), and supp− (p − q  ) ⊆ supp− (p − q). Repeated applications of (5.14) to (p, q) yield p , q  ∈ P , and an application of (5.14) to (p , q  ) gives p , q  ∈ P . [(5.15)⇒(SBS [Z])]: For p, q ∈ P , define a sequence (q (0) , q (1) , . . .) of integer points as follows:  q (0) = q,

q (k+1) =

p + q (k) 2

 (k = 0, 1, . . .).

Here, note that q (k) ∈ P (k = 0, 1, . . .). We see that (i) p(v) − q (k) (v) ∈ {0, 1} =⇒ q (k+1) (v) = q (k) (v), (ii) p(v) − q (k) (v) ≥ 2 =⇒ p(v) > q (k+1) (v) = q (k) (v) + 12 (p(v) − q (k) (v)) ≥ q (k) (v) + 1, (iii) p(v) − q (k) (v) ≤ −1 =⇒ p(v) ≤ q (k+1) (v) = q (k) (v) − 12 (q (k) (v) − p(v)) ≤ q (k) (v) − 1. It follows that there exists some positive integer N such that q (k) = q (N ) for any integer k ≥ N . Because of (i)–(iii) such a q (N ) is equal to (p − 1) ∨ (p ∧ q) and hence we have (p − 1) ∨ (p ∧ q) ∈ P . Replacing p with (p − 1) ∨ (p ∧ q) and repeating the above argument, we also have (p − 2 · 1) ∨ (p ∧ q) ∈ P . Repeating this argument (or more rigorously by induction), we have (p − α1) ∨ (p ∧ q) ∈ P for α ∈ Z+ . In particular, we have p ∧ q ∈ P . By symmetry we also have (p ∨ q) ∧ (q + α1) ∈ P for α ∈ Z+ and, in particular, p ∨ q ∈ P . Now, replacing p with p ∨ q in the above argument from the beginning, we have (p − α1) ∨ q ∈ P for α ∈ Z+ . By symmetry we also have p ∧ (q + α1) ∈ P for α ∈ Z+ .

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5.6. L-Convex Polyhedra

5.6

131

L-Convex Polyhedra

L-convex polyhedra are defined in section 5.3 as the convex hull of L-convex sets, and as such they are necessarily integral polyhedra. The concept of L-convexity, however, can also be defined for general (nonintegral) polyhedra. A nonempty polyhedron D ⊆ RV is defined to be an L-convex polyhedron if it satisfies (SBS[R]) (TRS[R])

p, q ∈ D =⇒ p ∨ q, p ∧ q ∈ D, p ∈ D =⇒ p + α1 ∈ D (∀ α ∈ R).

By an integral L-convex polyhedron we mean an L-convex polyhedron D such that D = D ∩ ZV . An integral polyhedron D is L-convex if and only if D ∩ ZV is an L-convex set. We denote by L0 [R] the set of L-convex polyhedra and by L0 [Z|R] the set of integral L-convex polyhedra. Let γ be a distance function and assume D(γ) = ∅. Then D(γ) is an Lconvex polyhedron. If, in addition, γ is integer valued, D(γ) is an integral L-convex polyhedron. The one-to-one correspondence of L-convex sets with distance functions (Theorem 5.5) is generalized as follows: ⇐⇒ D = D(γ) for γ ∈ T [R], (5.18)

D is an L-convex polyhedron

D is an integral L-convex polyhedron ⇐⇒ D = D(γ) for γ ∈ T [Z]. (5.19) The restriction of an L-convex polyhedron to a coordinate plane is called an L -convex polyhedron. A polyhedron P ⊆ RV is L -convex if and only if (SBS [R])

p, q ∈ P =⇒ (p − α1) ∨ q, p ∧ (q + α1) ∈ P

(∀ α ∈ R+ ).

We also have P ⊆ RV is an L -convex polyhedron ⇐⇒ P = P(γ, γˆ , γˇ )

(∃ γ, γˆ , γˇ ),

(5.20)

where P(γ, γˆ , γˇ ) is defined in (5.16) and γ˜ in (5.17) belongs to T [R]. We denote by L0 [R] and L0 [Z|R] the sets of L -convex polyhedra and integral L -convex polyhedra, respectively. An L-convex cone means a cone that is an L-convex polyhedron. We have ) ( * )  ) D is an L-convex cone ⇐⇒ D = cX χX ) cX ≥ 0 (X ∈ D \ {V }) ) X∈D

for some ring family D ⊆ 2V with V ∈ D,

(5.21)

as is proved in Note 8.10. An L-convex polyhedron is characterized as a polyhedron such that the tangent cone at each point is an L-convex cone (by (a) ⇔ (b) in Theorem 7.45).

Bibliographical Notes The concept of L-convex sets was introduced by Murota [140] and that of L -convex sets by Fujishige–Murota [68]. The polyhedron D(γ) associated with a distance

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Chapter 5. L-Convex Sets and Distance Functions

function is a well-studied object, appearing, e.g., in the dual of the trans-shipment problem. In particular, (5.5) is known as the maximum-separation minimum-route theorem (Theorem 21.1 of Iri [94]) or as the max tension min path theorem (section 6C of Rockafellar [178]). Theorem 5.5 (the description of L-convex sets by D(γ)) and Theorem 5.9 (separation of L-convex sets) are due to Murota [140]. Theorem 5.2 (hole-free property), Theorem 5.7 (convexity in intersection), and Theorem 5.8 (convexity in Minkowski sum) are by Murota [141]. Integral convexity of L-convex sets (Theorem 5.10) is due to Murota–Shioura [153]. The example in Note 5.11 is an adaptation of Example 3.11 of [153]. Nonintegral L-convex polyhedra in section 5.6 are considered in Murota–Shioura [152].

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Chapter 6

M-Convex Functions

M-convex functions form a class of well-behaved discrete convex functions. They are defined in terms of an exchange axiom and are characterized as functions obtained by piecing together M-convex sets in a consistent way or as collections of distance functions with some consistency. Fundamental properties of M-convex functions are established in this chapter, including the local optimality criterion for global optimality, the proximity theorem for minimizers, integral convexity, and extensibility to convex functions. Duality and conjugacy issues are treated in Chapter 8 and algorithms in Chapter 10.

6.1

M-Convex Functions and M -Convex Functions

We recall the definitions of M-convex functions and M -convex functions from section 1.4.2. A function f : ZV → R ∪ {+∞} with dom f = ∅ is said to be an M-convex function if it satisfies the following exchange axiom: (M-EXC[Z]) For x, y ∈ dom f and u ∈ supp+ (x − y), there exists v ∈ supp− (x − y) such that f (x) + f (y) ≥ f (x − χu + χv ) + f (y + χu − χv ).

(6.1)

Inequality (6.1) implicitly imposes the condition that x − χu + χv ∈ dom f and y + χu − χv ∈ dom f . With the use of the notation Δf (z; v, u) = f (z + χv − χu ) − f (z)

(z ∈ dom f ; u, v ∈ V ),

(6.2)

the exchange axiom (M-EXC[Z]) can be expressed alternatively as follows: (M-EXC [Z]) For x, y ∈ dom f , max

min

u∈supp+ (x−y) v∈supp− (x−y)

[Δf (x; v, u) + Δf (y; u, v)] ≤ 0, 133

(6.3)

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134

Chapter 6. M-Convex Functions

where the maximum and the minimum over an empty set are −∞ and +∞, respectively. We denote by M[Z → R] the set of M-convex functions and by M[Z → Z] the set of integer-valued M-convex functions. Proposition 6.1. The effective domain of an M-convex function is an M-convex set. Therefore, it lies on a hyperplane {x ∈ RV | x(V ) = r} for some integer r. Proof. It follows from (M-EXC[Z]) that B = dom f satisfies (B-EXC[Z]). Then the latter half follows from Proposition 4.1. Since the effective domain of an M-convex function f lies on a hyperplane, we may consider, instead of the function f in n = |V | variables, the projection f  of f along an arbitrarily chosen coordinate axis u0 ∈ V , where the projection f  is a function in n − 1 variables defined by f  (x ) = f (x0 , x )

for x0 = r − x (V  ) 

with the notation V  = V \ {u0 } and (x0 , x ) ∈ Z × ZV . A function derived from an M-convex function by such a projection is called an M -convex function. More formally, let 0 denote a new element not in V , and put V˜ = {0} ∪ V . A ˜ function f : ZV → R∪{+∞} is called M -convex if the function f˜ : ZV → R∪{+∞} defined by  f (x) if x0 = −x(V ) f˜(x0 , x) = (6.4) (x0 ∈ Z, x ∈ ZV ) +∞ otherwise is an M-convex function. We denote by M [Z → R] the set of M -convex functions and by M [Z → Z] the set of integer-valued M -convex functions. To characterize M -convex functions we introduce another exchange axiom: (M -EXC[Z]) For x, y ∈ dom f and u ∈ supp+ (x − y),  f (x) + f (y) ≥ min f (x − χu ) + f (y + χu ),  min

{f (x − χu + χv ) + f (y + χu − χv )} .

v∈supp− (x−y)

(6.5)

An alternative form of (M -EXC[Z]) using the notation (6.2) is as follows: (M -EXC [Z]) For x, y ∈ dom f , max

min

[Δf (x; v, u) + Δf (y; u, v)] ≤ 0,

u∈supp+ (x−y) v∈supp− (x−y)∪{0}

where, by convention, χ0 is the zero vector, and Δf (x; 0, u) = f (x − χu ) − f (x),

Δf (y; u, 0) = f (y + χu ) − f (y).

(6.6)

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6.2. Local Exchange Axiom

135

Theorem 6.2. For a function f : ZV → R ∪ {+∞} with dom f = ∅, we have f is M -convex ⇐⇒ f satisfies (M -EXC[Z]). Proof. (M-EXC[Z]) for f˜ in (6.4) is translated to conditions on f as follows: x(V ) > y(V ) ⇒ max+ u∈S

min

[Δf (x; v, u) + Δf (y; u, v)] ≤ 0,

(6.7)

v∈S − ∪{0}

x(V ) = y(V ) ⇒ max min [Δf (x; v, u) + Δf (y; u, v)] ≤ 0,

(6.8)

x(V ) < y(V ) ⇒

(6.9)

u∈S + v∈S −

max

min [Δf (x; v, u) + Δf (y; u, v)] ≤ 0,

u∈S + ∪{0} v∈S −

where S + = supp+ (x−y) and S − = supp− (x−y). As is easily seen, these conditions imply (6.5). The converse is shown in Note 6.6 in section 6.2. M -convex functions are conceptually equivalent to M-convex functions, but the class of M -convex functions is larger than that of M-convex functions. Theorem 6.3. An M-convex function is M -convex. Conversely, an M -convex function is M-convex if and only if the effective domain is contained in {x ∈ ZV | x(V ) = r} for some r ∈ Z. Proof. The first half follows from the obvious implication (M-EXC[Z])⇒(M EXC[Z]) and Theorem 6.2. The second half is from the equivalence of (M-EXC[Z]) and (M -EXC[Z]) under the condition on the effective domain. For ease of reference we summarize the relationship between M and M as Mn ⊂ Mn  Mn+1 ,

(6.10)

where Mn and Mn denote, respectively, the sets of M-convex functions and M convex functions in n variables, and the expression Mn  Mn+1 means a correspondence of their elements (functions) up to a translation of the effective domain along a coordinate axis, where (6.4) gives the correspondence under the normalization of r = 0. By the equivalence between M-convex functions and M -convex functions, all theorems stated for M-convex functions can be rephrased for M -convex functions, and vice versa. In this book we primarily work with M-convex functions, making explicit statements for M -convex functions when appropriate.

6.2

Local Exchange Axiom

There are a number of axioms equivalent to (M-EXC[Z]). We consider here a local exchange axiom: (M-EXCloc [Z]) For x, y ∈ dom f with ||x − y||1 = 4, there exist u ∈ supp+ (x − y) and v ∈ supp− (x − y) such that (6.1) holds true.

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Chapter 6. M-Convex Functions

On expressing y = x − χu1 − χu2 + χv1 + χv2 with u1 , u2 , v1 , v2 ∈ V and {u1 , u2 } ∩ {v1 , v2 } = ∅, we see that (M-EXCloc [Z]) is written as f (x − χu1 − χu2 + χv1 + χv2 ) − f (x) ≥ min[Δf (x; v1 , u1 ) + Δf (x; v2 , u2 ), Δf (x; v2 , u1 ) + Δf (x; v1 , u2 )]. (6.11) Theorem 6.4. If dom f is an M-convex set, then (M-EXC[Z]) ⇐⇒ (M-EXCloc [Z]). Proof. It suffices to show that (M-EXCloc [Z]) ⇒ (M-EXC[Z]). To prove this by contradiction, we assume that there exists a pair (x, y) for which (M-EXC[Z]) fails. That is, we assume the set of such pairs D = {(x, y) | x, y ∈ B, ∃ u∗ ∈ supp+ (x − y), ∀ v ∈ supp− (x − y) : Δf (x; v, u∗ ) + Δf (y; u∗ , v) > 0} is nonempty, where B = dom f . Take a pair (x, y) ∈ D with minimum ||x − y||1 , where ||x − y||1 > 4 by (M-EXCloc [Z]), and fix u∗ ∈ supp+ (x − y) appearing in the definition of D. For a fixed ε > 0, define p : V → R by ⎧ (v ∈ supp− (x − y), x − χu∗ + χv ∈ B), Δf (x; v, u∗ ) ⎪ ⎪ ⎨ −Δf (y; u∗ , v) + ε (v ∈ supp− (x − y), x − χu∗ + χv ∈ / B, p(v) = − χ ∈ B), y + χ ⎪ u∗ v ⎪ ⎩ 0 (otherwise). We use the notation Δfp (z; v, u) = Δf (z; v, u) + p(u) − p(v)

(z ∈ B, u, v ∈ V ).

Claim 1: Δfp (x; v, u∗ ) = 0 Δfp (y; u∗ , v) > 0

(v ∈ supp− (x − y), x − χu∗ + χv ∈ B), (v ∈ supp− (x − y)).

(6.12) (6.13)

(Proof of Claim 1) The equality (6.12) is obvious from the definition of p, and (6.13) can be seen as follows. If x−χu∗ +χv ∈ B, (6.13) follows from Δfp (x; v, u∗ ) = 0 (by (6.12)) and Δfp (x; v, u∗ ) + Δfp (y; u∗ , v) = Δf (x; v, u∗ ) + Δf (y; u∗ , v) > 0 / B, (6.13) follows from the fact that (by the definition of u∗ ). If x − χu∗ + χv ∈ Δfp (y; u∗ , v) = ε or +∞ depending on whether y + χu∗ − χv ∈ B or not. Claim 2: There exist u0 ∈ supp+ (x − y) and v0 ∈ supp− (x − y) such that y + χu0 − χv0 ∈ B, u∗ ∈ supp+ (x − (y + χu0 − χv0 )), and Δfp (y; u0 , v0 ) ≤ Δfp (y; u0 , v)

(v ∈ supp− (x − y)).

(6.14)

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137

(Proof of Claim 2) Put u0 = u∗ if x(u∗ ) ≥ y(u∗ ) + 2; otherwise, take any u0 ∈ supp+ (x − y) \ {u∗ }, which is possible by ||x − y||1 > 4. By (B-EXC[Z]) there exists v ∈ supp− (x − y) such that y + χu0 − χv ∈ B. Let v0 ∈ supp− (x − y) be such a v that minimizes Δfp (y; u0 , v). Then we have (6.14). Claim 3: (x, y  ) ∈ D for y  = y + χu0 − χv0 . (Proof of Claim 3) It suffices to show Δfp (x; v, u∗ ) + Δfp (y  ; u∗ , v) > 0

(v ∈ supp− (x − y  )).

(6.15)

We may assume x − χu∗ + χv ∈ B, since otherwise Δfp (x; v, u∗ ) = +∞. Then Δfp (x; v, u∗ ) = 0 by (6.12) and Δfp (y  ; u∗ , v) = f [−p](y + χu0 + χu∗ − χv0 − χv ) − f [−p](y + χu0 − χv0 ) ≥ min [Δfp (y; u0 , v0 ) + Δfp (y; u∗ , v), Δfp (y; u0 , v) + Δfp (y; u∗ , v0 )] −Δfp (y; u0 , v0 ) > min [Δfp (y; u0 , v0 ), Δfp (y; u0 , v)] − Δfp (y; u0 , v0 ) = 0 by (M-EXCloc [Z]), (6.13), and (6.14). This establishes Claim 3. Since ||x − y  || = ||x − y|| − 2, Claim 3 is a contradiction to the choice of (x, y). Therefore, D must be an empty set. As a corollary to Theorem 6.4 we see that (M-EXC[Z]) is equivalent to a weak exchange axiom: (M-EXCw [Z]) For distinct x, y ∈ dom f , there exist u ∈ supp+ (x − y) and v ∈ supp− (x − y) such that (6.1) holds true. Note the difference in the two axioms: “∀ u, ∃ v” in (M-EXC[Z]) and “∃ u, ∃ v” in (M-EXCw [Z]). Theorem 6.5. (M-EXC[Z]) ⇐⇒ (M-EXCw [Z]). Proof. It suffices to show ⇐ when dom f =  ∅. (M-EXCw [Z]) for f implies (BEXCw [Z]) for dom f , and therefore, dom f is an M-convex set by Theorem 4.3. Then Theorem 6.4 establishes the claim.

Note 6.6. The proof of Theorem 6.2 is completed here. It remains to show that (M -EXC[Z]) for f implies (M-EXC[Z]) for f˜ defined by (6.4). First, dom f˜ is an M-convex set, since (M -EXC[Z]) for f implies (B -EXC[Z]) for dom f and dom f is the projection of dom f˜ (see section 4.7 and (4.35) in particular). By Theorem 6.4, (M-EXC[Z]) is equivalent to (M-EXCloc [Z]), and therefore, it suffices to show (6.7), (6.8), and (6.9) for x, y such that x − y = χu1 + χu2 − χv1 − χv2 with {u1 , u2 , v1 , v2 } ⊆ V ∪ {0} and {u1 , u2 } ∩ {v1 , v2 } = ∅. Since (6.7) is obvious from (6.5), it remains to show (6.8) for four cases:

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(a1) x − y = χu1 + χu2 − χv1 − χv2 , (a2) x − y = 2χu1 − 2χv1 , (a3) x − y = 2χu1 − χv1 − χv2 , (a4) x − y = χu1 + χu2 − 2χv1 , and (6.9) for four cases: (b1) x − y = χu1 − χv1 − χv2 , (b2) x − y = −2χv1 , (b3) x − y = −χv1 − χv2 , (b4) x − y = χu1 − 2χv1 , where u1 , u2 , v1 , v2 are all distinct. We deal with (a1) and (b4) below; the other cases are left to the reader. Case (a1): We abbreviate z = (x ∧ y) + α1 χu1 + α2 χu2 + β1 χv1 + β2 χv2 to (α1 α2 β1 β2 ); for instance, x = (1100) and y = (0011). We are to derive f (1100) + f (0011) ≥ min[f (0110) + f (1001), f (1010) + f (0101)].

(6.16)

By (6.5) for u = u1 , we have (6.16) or f (1100) + f (0011) ≥ f (0100) + f (1011).

(6.17)

By (6.5) for u = u2 , we have (6.16) or f (1100) + f (0011) ≥ f (1000) + f (0111).

(6.18)

Furthermore, by (6.5) we have f (0100) + f (0111) ≥ f (0110) + f (0101),

(6.19)

f (1000) + f (1011) ≥ f (1010) + f (1001).

(6.20)

Adding (6.17), (6.18), (6.19), and (6.20) yields 2[f (1100) + f (0011)] ≥ [f (0110) + f (1001)] + [f (1010) + f (0101)], which implies (6.16). Case (b4): We abbreviate z = (x ∧ y) + α1 χu1 + β1 χv1 to (α1 β1 ); for instance, x = (10) and y = (02). We are to show f (10) + f (02) ≥ f (01) + f (11). This, however, is derived from (6.5) for (02), (10), and u = v1 .

6.3

Examples

We have already seen M-convexity in network flows and in matroids (section 2.2, section 2.4). In this section we see some other examples of M-convex functions, such as linear functions, quadratic functions, and separable convex functions. We start by recalling the following facts from Proposition 6.1. Proposition 6.7. (1) The effective domain of an M-convex function is an M-convex set. (2) The effective domain of an M -convex function is an M -convex set.

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6.3. Examples Linear functions

139 A linear (or affine) function45 f (x) = α + p, x

(x ∈ dom f ),

(6.21)

with p ∈ Rn and α ∈ R, is M-convex or M -convex according as dom f is M-convex or M -convex. The inequalities (6.1) in (M-EXC[Z]) and (6.5) in (M -EXC[Z]) are satisfied with equality. Quadratic functions

A separable quadratic function f (x) =

n 

ai x(i)2

(x ∈ dom f ),

(6.22)

i=1

with ai ∈ R+ (i = 1, . . . , n), is M-convex if dom f is an M-convex set. A quadratic function of the form f (x) =

n  i=1

2

ai x(i) + b



x(i)x(j)

(x ∈ Zn ),

(6.23)

i y(X),

(6.38)

u∈ / X, v ∈ X, X ∈ T =⇒ x(X) < y(X).

(6.39)

If x(X) > y(X) for all X ∈ T containing u, we can take v = 0 to meet (6.38) and (6.39). Otherwise, let X0 be the unique minimal element of T such that u ∈ X0 and x(X0 ) ≤ y(X0 ).5By the minimality of X0 and (6.37) we have (i) ∃ v ∈ X0 \ Y ∈T (X0 ) Y such that x(v) < y(v) or (ii) ∃ X1 ∈ T (X0 ) such that x(X1 ) < y(X1 ). In case (i), this v is5valid for (6.38) and (6.39). In case (ii), from (6.37) follows (i) ∃ v ∈ X1 \ Y ∈T (X1 ) Y such that x(v) < y(v) or (ii) ∃ X2 ∈ T (X1 ) such that x(X2 ) < y(X2 ). Repeating this argument we eventually arrive at case (i). Note 6.12. In section 2.1 we considered M -convex quadratic functions in real variables, whereas we have investigated functions in integer variables in this section; namely, Rn → R in section 2.1 and Zn → R here. Both are characterized by exchange properties; the former by (M -EXC[R]) and the latter by (M -EXC[Z]). One of the main results of section 2.1, Theorem 2.12, says that a positive-definite symmetric matrix belongs to the class L−1 of (2.19) if and only if the associated quadratic form satisfies (M -EXC[R]). This statement, however, does not carry over to the discrete setting. For instance, consider ⎡ ⎤ ⎡ ⎤ 16 11 13 5 −1 −2 1 ⎣ 11 21 17 ⎦ , A−1 = ⎣ −1 5 −3 ⎦ , A= 43 13 17 24 −2 −3 5 where A ∈ L−1 . The associated quadratic form f (x) = 12 x Ax satisfies (M EXC[R]) as a function f : R3 → R in real variables, but does not meet (M EXC[Z]) when viewed as f : Z3 → R in integer variables. This phenomenon seems to be indicative of the subtleties inherent in discreteness. See Note 8.13 for the conjugate of M -convex quadratic functions.

6.4

Basic Operations

Basic operations on M-convex functions are presented here, whereas a most important operation, transformation by networks, is treated later in section 9.6.

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6.4. Basic Operations

143

First we introduce some operations on a function f : ZV → R ∪ {+∞} in general. For a subset U ⊆ V , the restriction, the projection, and the aggregation of f to U are functions fU : ZU → R ∪ {+∞}, f U : ZU → R ∪ {±∞}, and f U∗ : ZU × Z → R ∪ {±∞} defined respectively by fU (y) = f (y, 0V \U ) f

(y ∈ ZU ),

f U (y) = inf{f (y, z) | z ∈ ZV \U } (y ∈ ZU ), (y, w) = inf{f (y, z) | z(V \ U ) = w, z ∈ ZV \U }

U∗

(6.40) (y ∈ ZU , w ∈ Z),

(6.41) (6.42)

where 0V \U means the zero vector in ZV \U . For a pair of functions fi : ZV → R ∪ {+∞} (i = 1, 2), the integer infimal convolution is a function f1 2Z f2 : ZV → R ∪ {±∞} defined by (f1 2Z f2 )(x) = inf{f1 (x1 ) + f2 (x2 ) | x = x1 + x2 , x1 , x2 ∈ ZV } (x ∈ ZV ). (6.43) Provided that f1 2Z f2 is away from the value of −∞, we have dom (f1 2Z f2 ) = dom f1 + dom f2 ,

(6.44)

where the right-hand side means the discrete Minkowski sum (3.52). The projection f U can be represented as f U = (f 2Z δUˆ )U , (6.45) which says that f U coincides with the restriction to U of the integer infimal conˆ = {x ∈ ZV | x(v) = 0 (v ∈ U )}. volution of f with the indicator function δUˆ of U We continue to use the notation f [−p] for p ∈ RV defined in (3.69) and f[a,b] for an integer interval [a, b] defined in (3.55). M-convex functions admit the following operations. Theorem 6.13. Let f, f1 , f2 ∈ M[Z → R] be M-convex functions. (1) For λ ∈ R++ , λf is M-convex. (2) For a ∈ ZV , f (a − x) and f (a + x) are M-convex in x. (3) For p ∈ RV , f [−p] is M-convex. (4) For ϕv ∈ C[Z → R] (v ∈ V ),  f˜(x) = f (x) + ϕv (x(v)) (x ∈ ZV )

(6.46)

v∈V

is M-convex provided dom f˜ = ∅. (5) For a, b ∈ (Z ∪ {±∞})V , the restriction f[a,b] to the integer interval [a, b] is M-convex provided dom f[a,b] = ∅. (6) For U ⊆ V , the restriction fU is M-convex provided dom fU = ∅. (7) For U ⊆ V , the aggregation f U∗ is M-convex provided f U∗ > −∞. (8) The integer infimal convolution f˜ = f1 2Z f2 is M-convex provided f˜ > −∞. Proof. (1), (2), (5), and (6) are obvious and (3) is a special case of (4).

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(4) For x, y ∈ dom f˜ ⊆ dom f and u ∈ supp+ (x − y), use (M-EXC[Z]) for f to obtain v ∈ supp− (x − y) satisfying (6.1) for f . Then f˜(x − χu + χv ) + f˜(y + χu − χv ) − f˜(x) − f˜(y) = [f (x − χu + χv ) + f (y + χu − χv ) − f (x) − f (y)] + [ϕu (x(u) − 1) + ϕu (y(u) + 1) − ϕu (x(u)) − ϕu (y(u))] + [ϕv (x(v) + 1) + ϕv (y(v) − 1) − ϕv (x(v)) − ϕv (y(v))] ≤ 0. (7) We show this in Note 9.29 using transformation by a network. (8) We show this in Note 9.30 using transformation by a network. As is easily seen, the converse of Theorem 6.13 (5) is also true. Proposition 6.14. For a function f : ZV → R ∪ {+∞}, we have f is M-convex ⇐⇒ f[a,b] is M-convex for any a, b ∈ ZV with dom f[a,b] = ∅. The operations in Theorem 6.13 are also valid for M -convex functions. In addition, the projection is allowed for M -convex functions. Theorem 6.15. Let f, f1 , f2 ∈ M [Z → R] be M -convex functions. (1) Operations (1)–(8) of Theorem 6.13 are valid for M -convex functions. (2) For U ⊆ V , the projection f U is M -convex provided f U > −∞. Proof. (2) This follows from (6.45) as well as the M -versions of (6) and (8) of Theorem 6.13.

Note 6.16. The sum of two M-convex functions is not necessarily M-convex. For example, recall the M-convex sets B1 and B2 in Note 4.25 such that B1 ∩ B2 is not M-convex. Their indicator functions are M-convex, but their sum, which is the indicator function of B1 ∩ B2 , is not M-convex. The sum of two M-convex functions is studied under the name M2 -convex function in section 8.3. A similar argument applies to the sum of two M -convex functions. Note 6.17. The proviso f U∗ > −∞ in Theorem 6.13 (7) can be weakened to f U∗ (x0 ) > −∞ for some x0 . A similar weakening holds for f˜ > −∞ in Theorem 6.13 (8) and f U > −∞ in Theorem 6.15 (2). Note 6.18. For a function f : ZV → R ∪ {+∞} and a positive integer α, we define a function f α : ZV → R ∪ {+∞} by f α (x) =

1 f (αx) α

(x ∈ ZV ).

(6.47)

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6.5. Supermodularity

145 6

f (x) αf α ( αx )

−2 −1

0

1

2

3

4 x

Figure 6.1. Scaling f α for α = 2.

This is called a scaling in the domain or a domain scaling. If α = 2, for instance, this amounts to considering the function values only on vectors of even integers (see Fig. 6.1). Scaling is one of the common techniques used in designing efficient algorithms—this is particularly true of network flow algorithms (see Ahuja–Magnanti–Orlin [1]). M-convexity (or M -convexity) is not preserved under scaling. For example, the indicator function f of an M-convex set {c1 (1, 0, −1, 0) + c2 (1, 0, 0, −1) + c3 (0, 1, −1, 0) + c4 (0, 1, 0, −1) | ci ∈ {0, 1}} is M-convex, but f α for α = 2 is not, because it is the indicator function of {(0, 0, 0, 0), (1, 1, −1, −1)}, which is not M-convex. Nevertheless, scaling an Mconvex function is useful in designing efficient algorithms, as we will see in section 10.1 as well as in Theorem 6.37 (a proximity theorem for M-convex functions). It is worth mentioning that some subclasses of M-convex functions are closed under the scaling operation; linear, quadratic, separable, and laminar M-convex functions form such subclasses. See Proposition 10.41 for a type of scaling operation for M-convex functions.

6.5

Supermodularity



M -convex functions are supermodular on the integer lattice. Theorem 6.19. An M -convex function f ∈ M [Z → R] is supermodular; i.e., f (x) + f (y) ≤ f (x ∨ y) + f (x ∧ y)

(x, y ∈ ZV ).

(6.48)

Proof. For x ∈ ZV , (M -EXC[Z]) applied to (x + χu + χv , x) yields f (x + χu ) + f (x + χv ) ≤ f (x + χu + χv ) + f (x)

(u, v ∈ V, u = v).

(6.49)

We prove (6.48) by induction on ||x − y||1 , where we may assume supp+ (x − y) = ∅ and supp− (x − y) = ∅. By (6.49), (6.48) is true if ||x − y||1 ≤ 2. For x, y with

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||x − y||1 ≥ 3, we may assume x ∨ y, x ∧ y ∈ dom f and also {x(u) − y(u) | u ∈ supp+ (x − y)} ≥ 2, by symmetry. Take u ∈ supp+ (x − y) and put x = (x ∧ y) + χu and y  = y + χu . Since dom f is an M -convex set, it includes the integer interval [x ∧ y, x ∨ y]Z and, in particular, x , y  ∈ dom f . By ||x − y||1 ≤ ||x − y||1 − 1 and ||x − y  ||1 = ||x − y||1 − 1, the induction hypothesis yields f (y) − f (x ∧ y) ≤ f (y + χu ) − f ((x ∧ y) + χu ) ≤ f (x ∨ y) − f (x), which shows (6.48).

Example 6.20. The converse of Theorem 6.19 is not true. For instance, a function f : Z3 → R ∪ {+∞} defined by dom f = {0, 1}3 and f (1, 1, 1) = 2, f (1, 1, 0) = f (1, 0, 1) = 1, f (0, 0, 0) = f (1, 0, 0) = f (0, 1, 0) = f (0, 0, 1) = f (0, 1, 1) = 0 is supermodular and not M -convex; (M -EXC[Z]) fails for x = (0, 1, 1), y = (1, 0, 0), and u = 2. Note 6.21. We have repeatedly said that submodularity corresponds to convexity (in section 4.5, in particular). Theorem 6.19 says, however, that M -concave functions are submodular. Though somewhat annoying, this is not a contradiction, but provides a better understanding of the fact that motivated the analogy of submodularity to concavity in the 1970s. The fact is that, for a univariate concave function h, the set function ρ defined by ρ(X) = h(|X|) for X ⊆ V is submodular (Edmonds [44], Lov´asz [123]). A possible understanding based on Theorem 6.19 is as follows: ρ is an M -concave function, viewed as a function on {0, 1}V , and, therefore, it is submodular. Recall also section 2.3.1 for the issue of convexity vs. submodularity.

Note 6.22. The supermodular inequality (6.48) is void for an M-convex function f because x ∨ y, x ∧ y ∈ dom f occurs only when x = y. For an M-convex function f , the property corresponding to (6.49) is expressed as f (x + (χu − χw )) + f (x + (χv − χw )) ≤ f (x + (χu − χw ) + (χv − χw )) + f (x), where u, v, w are distinct elements of V and x ∈ ZV .

6.6

Descent Directions

One of the most conspicuous features of an M-convex function f is that it has a prescribed set of possible descent directions in the sense that x, y ∈ dom f , f (x) > f (y) =⇒ f (x) >

min

min

u∈supp+ (x−y) v∈supp− (x−y)

f (x − χu + χv ).

This is an exemplar of what we understand as discreteness in direction.

(6.50)

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6.6. Descent Directions

147

Proposition 6.23. An M-convex function f ∈ M[Z → R] satisfies (6.50). Proof. By (M-EXC[Z]) there exist u1 ∈ supp+ (x − y) and v1 ∈ supp− (x − y) such that f (y) ≥ [f (x − χu1 + χv1 ) − f (x)] + f (y2 ), where y2 = y + χu1 − χv1 . By (M-EXC[Z]) applied to (x, y2 ), there exist u2 ∈ supp+ (x − y2 ) and v2 ∈ supp− (x − y2 ) such that f (y2 ) ≥ [f (x − χu2 + χv2 ) − f (x)] + f (y3 ), where y3 = y2 +χu2 −χv2 = y+χu1 +χu2 −χv1 −χv2 . Repeating m this m = ||x−y||1 /2 times, we obtain (ui , vi ) (i = 1, . . . , m) such that y = x − i=1 (χui − χvi ) and f (x) > f (y) ≥ f (x) +

m 

[f (x − χui + χvi ) − f (x)].

i=1

Therefore, f (x − χui + χvi ) − f (x) < 0 for some i. The property (6.50) is essential for M-convexity. For an M-convex function f and any p ∈ RV , f [p] is again M-convex, and, therefore, f satisfies the following property: (M-SI[Z]) For p ∈ RV and x, y ∈ dom f with f [p](x) > f [p](y), f [p](x) >

min

min

u∈supp+ (x−y) v∈supp− (x−y)

f [p](x − χu + χv ).

(6.51)

As the M -version we consider the following: (M -SI[Z]) For p ∈ RV and x, y ∈ dom f with f [p](x) > f [p](y), f [p](x) >

min

min

u∈supp+ (x−y)∪{0} v∈supp− (x−y)∪{0}

f [p](x − χu + χv ),

(6.52)

where χ0 = 0, as usual. Theorem 6.24. Let f : ZV → R ∪ {+∞} be a function with dom f = ∅. (1) f is an M-convex function ⇐⇒ f satisfies (M-SI[Z]). (2) f is an M -convex function ⇐⇒ f satisfies (M -SI[Z]). Proof. It suffices to prove (1). The implication ⇒ is immediate from Theorem 6.13 (3) and Proposition 6.23. The converse ⇐ follows from Claims 1 and 2 below by Theorem 6.4. Claim 1: B = dom f is an M-convex set. (Proof of Claim 1) For x, y ∈ B and u ∈ supp+ (x − y), take a sufficiently large M > 0 and define p : V → R by ⎧ ⎨ M 2 (v = u), M (v ∈ supp− (x − y)), p(v) = ⎩ 0 (otherwise).

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Then f [p](x) > f [p](y), and by (M-SI[Z]) there exist w ∈ supp+ (x − y) and v ∈ supp− (x − y) such that f [p](x) − f [p](x − χw + χv ) = f (x) − f (x − χw + χv ) + p(w) − M > 0. This is possible only if w = u, which shows (B-EXC− [Z]) for B. Then B is an M-convex set by Theorem 4.3. Claim 2: f satisfies the local exchange axiom (M-EXCloc [Z]). (Proof of Claim 2) Take x, y ∈ B with ||x − y||1 = 4 and put y = x − χu1 − χu2 + χv1 + χv2 with u1 , u2 , v1 , v2 ∈ V and {u1 , u2 } ∩ {v1 , v2 } = ∅. In the following we assume u1 = u2 and v1 = v2 (the other cases can be treated similarly). Consider a bipartite graph G = (V + , V − ; E) with vertex bipartition V + = {u1 , u2 }, V − = {v1 , v2 } and arc set E = {(ui , vj ) | Δf (x; vj , ui ) < +∞ (i, j = 1, 2)}. The graph G has a perfect matching as a consequence of (B-EXC[Z]) for B. We think of Δf (x; vj , ui ) as the weight of arc (ui , vj ) and apply Proposition 3.14 to obtain p : V → R such that Δf (x; vj , ui ) ≥ p(ui ) − p(vj )

(i, j = 1, 2),

(6.53)

and p(u1 ) + p(u2 ) − p(v1 ) − p(v2 ) is equal to the right-hand side of (6.11) (and p(v) = 0 for v ∈ V \ {u1 , u2 , v1 , v2 }). Failure of inequality (6.11) would imply f [p](x) ≤ f [p](x − χui + χvj )

f [p](x) > f [p](y),

(i, j = 1, 2),

a contradiction to (M-SI[Z]). The proof of Proposition 6.23 shows the following. Proposition 6.25. For an M-convex function f ∈ M[Z → R] and x, y ∈ dom f , we have f (y) ≥ f (x) + fˇ(x, y), (6.54) where fˇ(x, y) = inf λ

⎧ ⎨  ⎩

u,v∈V

 u,v∈V

6.7

) ) ) λuv [f (x − χu + χv ) − f (x)])) ) λuv (χv − χu ) = y − x, λuv ∈ Z+

⎫ ⎬ (u, v ∈ V ) . (6.55) ⎭

Minimizers

Global optimality for an M-convex function is characterized by local optimality. Theorem 6.26 (M-optimality criterion). (1) For an M-convex function f ∈ M[Z → R] and x ∈ dom f , we have f (x) ≤ f (y) (∀ y ∈ ZV ) ⇐⇒ f (x) ≤ f (x − χu + χv ) (∀ u, v ∈ V ).

(6.56)

(2) For an M -convex function f ∈ M [Z → R] and x ∈ dom f , we have  f (x) ≤ f (x − χu + χv ) (∀ u, v ∈ V ), V f (x) ≤ f (y) (∀ y ∈ Z ) ⇐⇒ (6.57) f (x) ≤ f (x ± χv ) (∀ v ∈ V ).

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149

e

T

e

Figure 6.2. Minimum spanning tree problem. Proof. It suffices to prove ⇐ in (1), but this follows from Proposition 6.23.

Example 6.27. The minimum spanning tree problem serves as a canonical example to illustrate the M-optimality criterion. Let G = (U, E) be a graph with vertex set U and arc set E. A set T of arcs is called a spanning tree if it forms a connected subgraph that contains no circuit and covers all the vertices. The minimum spanning tree problem is to find a spanning tree T that has the minimum weight with respect to a given weight w : E → R, where the weight of T is defined as e∈T w(e). It is well known that a spanning tree T has the minimum weight if and only if w(e) ≤ w(e ) for any e ∈ T and e ∈ E \ T such that T − e + e is a spanning tree (see Fig. 6.2). This well-known optimality criterion is a special case of Theorem 6.26 (1) applied to an M-convex function f : ZE → R ∪ {+∞} defined by  f (x) =

e∈T

+∞

w(e)

(x = χT , T is a spanning tree), (otherwise).

Note that, for a spanning tree T and arcs e ∈ T and e ∈ E \ T , we have f (χT ) ≤ f (χT − χe + χe ) if and only if (i) w(e) ≤ w(e ) or (ii) T − e + e is not a spanning tree. In this connection it is noted that T − e + e is a spanning tree if and only if e belongs to the unique circuit contained in T ∪ {e }, called the fundamental circuit with respect to (T, e ). Theorem 6.26 above shows how to verify the optimality of a given point with O(n2 ) function evaluations. The next theorem suggests how to find a minimizer. Stating that a given point can be easily separated from some minimizer, it serves as a basis of the domain reduction algorithm for M-convex function minimization, to be explained in section 10.1.3. Theorem 6.28 (M-minimizer cut). Let f : ZV → R ∪ {+∞} be an M-convex function with arg min f = ∅. (1) For x ∈ dom f and v ∈ V , let u ∈ V be such that f (x − χu + χv ) = min f (x − χs + χv ). s∈V

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Then there exists x∗ ∈ arg min f with x∗ (u) ≤ x(u) − 1 + χv (u). (2) For x ∈ dom f and u ∈ V , let v ∈ V be such that f (x − χu + χv ) = min f (x − χu + χt ). t∈V

Then there exists x∗ ∈ arg min f with x∗ (v) ≥ x(v) − χu (v) + 1. (3) For x ∈ dom f \ arg min f , let u, v ∈ V be such that f (x − χu + χv ) = min f (x − χs + χt ). s,t∈V

Then there exists x∗ ∈ arg min f with x∗ (u) ≤ x(u) − 1,

x∗ (v) ≥ x(v) + 1.

Proof. (1) Put x = x − χu + χv . Assume, to the contrary, that there is no x∗ ∈ arg min f with x∗ (u) ≤ x (u). Let x∗ be an element of arg min f with x∗ (u) being minimum. Then we have x∗ (u) > x (u). By applying (M-EXC[Z]) to x∗ , x , and u we obtain some w ∈ supp− (x∗ − x ) such that if Δf (x∗ ; w, u) > 0 then Δf (x ; u, w) < 0. Since Δf (x∗ ; w, u) > 0 by the choice of x∗ , we have f (x ) > f (x + χu − χw ) = f (x − χw + χv ), a contradiction to the property of u. (2) The proof is similar to that for (1). (3) Put x = x − χu + χv (= x). By (1) there exists x∗ ∈ arg min f such that x∗ (u) ≤ x (u); we assume that x∗ maximizes x∗ (v) among all such vectors. If x∗ (v) ≥ x (v) is not satisfied, (M-EXC[Z]) applies to x , x∗ , and v to yield some w ∈ supp− (x − x∗ ) satisfying (a) Δf (x ; w, v) < 0, (b) Δf (x∗ ; v, w) < 0, or (c) Δf (x ; w, v) = Δf (x∗ ; v, w) = 0. We have Δf (x ; w, v) ≥ 0 by x − χv + χw = x − χu +χw and the choice of u and v. We also have Δf (x∗ ; v, w) ≥ 0 by x∗ ∈ arg min f . Therefore, we have (c), which implies x∗ + χv − χw ∈ arg min f , a contradiction to the choice of x∗ . The minimizers of an M-convex function form an M-convex set, a property that is essential for a function to be M-convex. Proposition 6.29. For an M-convex function f ∈ M[Z → R], arg min f is an M-convex set if it is not empty. Proof. For x, y ∈ arg min f , we have x − χu + χv , y + χu − χv ∈ arg min f in (6.1). This shows that arg min f satisfies (B-EXC[Z]). The following theorem reveals that M-convex functions are characterized as functions obtained by piecing together M-convex sets in a consistent way. This

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shows how the concept of M-convex functions can be defined from that of Mconvex sets. Theorem 6.30. Let f : ZV → R ∪ {+∞} be a function with a bounded nonempty effective domain. (1) f is M-convex ⇐⇒ arg min f [−p] is an M-convex set for each p ∈ RV . (2) f is M -convex ⇐⇒ arg min f [−p] is an M -convex set for each p ∈ RV . Proof. It suffices to prove (1). The implication ⇒ is immediate from Theorem 6.13 (3) and Proposition 6.29. For ⇐, it suffices, by Theorem 6.4, to show that B = dom f is an M-convex set and f satisfies the local exchange axiom (M-EXCloc [Z]). Claim 1: B is an M-convex set. (Proof of Claim 1) Put Bp = arg min f [−p] for each p. Then we have B = 5 B for the convex hulls of B and Bp . For x, y ∈ B, there exists p such that y ∈ Bp p p and z ≡ tx + (1 − t)y ∈ Bp for some t > 0. It follows from (B-EXC+ [R]) of Bp that, for u ∈ supp+ (x − y) = supp+ (z − y), there exists v ∈ supp− (z − y) = supp− (x − y) such that y + α(χu − χv ) ∈ Bp ⊆ B for all sufficiently small α > 0. This shows (B-EXC+ [R]) for B. Therefore, B is an M-convex set. For (M-EXCloc [Z]), take x, y ∈ B with ||x−y||1 = 4. Let f : RV → R∪{+∞} be the convex closure of f , where it is noted that f is not assumed to be convex extensible. Let p ∈ RV be a subgradient of f at c = (x + y)/2 ∈ RV . We have c ∈ arg min f [−p] = Bp , where Bp is an integral M-convex polyhedron. Hence, the intersection of Bp with the interval I = [x ∧ y, x ∨ y]R is an integral M-convex polyhedron, in which c is contained. This means that c can be represented as a convex combination of some integral vectors, say, z1 , . . . , zm ∈ (I∩Bp )∩ZV = I∩Bp : c=

m 

λk zk ,

z k ∈ I ∩ Bp

(k = 1, . . . , m),

(6.58)

k=1

where m k=1 λk = 1 and λk > 0 (k = 1, . . . , m). Since ||x−y||1 = 4, we have y = x−χv1 −χv2 +χv3 +χv4 for some v1 , v2 , v3 , v4 ∈ V with {v1 , v2 } ∩ {v3 , v4 } = ∅. In the following we assume that v1 , v2 , v3 , and v4 are all distinct (the other cases can be treated similarly). Noting that any element z of I ∩ Bp can be represented as z = (x ∧ y) + χvi + χvj (i = j), we consider an undirected graph G = (V0 , E0 ) with vertex set V0 = {v1 , v2 , v3 , v4 } and edge set E0 = {{vi , vj } | zk = (x ∧ y) + χvi + χvj , k = 1, . . . , m}. Claim 2: G has a perfect matching (of size 2). (Proof of Claim 2) For each i (1 ≤ i ≤ 4), we have c(vi ) − (x ∧ y)(vi ) = 1/2, whereas zk (vi ) − (x ∧ y)(vi ) ∈ {0, 1} for all k in (6.58). Hence, for each i, there exist k1 and k0 such that zk1 (vi ) − (x ∧ y)(vi ) = 1,

zk0 (vi ) − (x ∧ y)(vi ) = 0.

Translating this into G, we see that for each vertex vi there is an edge that covers (is incident to) vi and also there is another edge that avoids (is not incident to) vi . This condition implies the existence of a perfect matching in G.

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Finally we derive (M-EXCloc [Z]) from Claim 2. We divide into two cases. (i) If {{v1 , v2 }, {v3 , v4 }} ⊆ E0 , both x and y appear among the zk ’s, and hence x, y ∈ Bp . By (B-EXC[Z]) for Bp , we have x − χvi + χvj ∈ Bp and y + χvi − χvj ∈ Bp for some i ∈ {1, 2} and j ∈ {3, 4}. Hence, f [−p](x) = f [−p](y) = f [−p](x − χvi + χvj ) = f [−p](y + χvi − χvj ), which shows (6.11) with equality. (ii) If {{v1 , v2 }, {v3 , v4 }} ⊆ E0 , it follows from Claim 2 that {{v1 , vi }, {v2 , vj }} ⊆ E0 for some i, j with {i, j} = {3, 4}. Then (x ∧ y) + χv1 + χvi = x − χv2 + χvi ,

(x ∧ y) + χv2 + χvj = y + χv2 − χvi

both belong to Bp ; i.e., f [−p](x − χv2 + χvi ) = f [−p](y + χv2 − χvi ) = min f [−p]. Hence, f [−p](x) + f [−p](y) ≥ f [−p](x − χv2 + χvi ) + f [−p](y + χv2 − χvi ), which establishes (6.11).

Note 6.31. The boundedness assumption on dom f in Theorem 6.30 is not restrictive substantially, since we know from Proposition 6.14 that f is M-convex if and only if its restriction f[a,b] to every bounded integer interval [a, b] is M-convex (as long as dom f[a,b] = ∅). On the other hand, the boundedness assumption seems inevitable. The function ⎧ (x = 0), ⎨ 0 1 (x = 0, x(1) + x(2) = 0), f (x) = ⎩ +∞ (otherwise) in x = (x(1), x(2)) ∈ Z2 is not M-convex, but, for each p ∈ RV , arg min f [−p] is equal to {0} (an M-convex set) if it is not empty.

6.8

Gross Substitutes Property

In the previous section we saw that the minimizers of an M-convex function f form an M-convex set for a fixed p ∈ RV . We investigate here how the minimizers of f [p] change with the variation of p. The term gross substitutes stems from an economic interpretation, where p represents the price vector; some background in mathematical economics will be given in section 11.3. We first observe a general phenomenon, independent of M-convexity, in the variation of minimizers. Let f : ZV → R ∪ {+∞} be any function and assume x ∈ arg min f [p] and y ∈ arg min f [q] for p, q ∈ RV . It follows from f [p](y) ≥ f [p](x) and f [q](x) ≥ f [q](y) that q − p, x − y = (f [p](y) − f [p](x)) + (f [q](x) − f [q](y)) ≥ 0.

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153

A particular case of this inequality with q = p + αχu for u ∈ V and α > 0 yields y(u) ≤ x(u). Namely, we have x ∈ arg min f [p], p ∈ RV , u ∈ V, α > 0, arg min f [p + αχu ] = ∅ =⇒ ∀ y ∈ arg min f [p + αχu ] : y(u) ≤ x(u).

(6.59)

This is a well-known phenomenon valid for any function f , a kind of monotonicity in the variation of minimizers. Note that nothing is claimed here about the other components x(v) and y(v) with v = u. The gross substitutes property that we consider in this section is concerned with the variation of other components. In contrast to (6.59) we introduce a condition on f : x ∈ arg min f [p], p ∈ RV , u ∈ V, α > 0, arg min f [p + αχu ] = ∅ =⇒ ∃ y ∈ arg min f [p + αχu ] : y(v) ≥ x(v) (∀ v ∈ V \ {u}).

(6.60)

Obviously, this condition is equivalent to the following: (M-GS[Z]) If x ∈ arg min f [p], p ≤ q, and arg min f [q] = ∅, there exists y ∈ arg min f [q] such that y(v) ≥ x(v) for all v ∈ V with p(v) = q(v). It should be clear that the inequality p ≤ q above means p(v) ≤ q(v) (∀ v ∈ V ). Proposition 6.32. An M-convex function f ∈ M[Z → R] satisfies (M-GS[Z]). Proof. For x ∈ arg min f [p] and p ≤ q, let y be an element of arg min f [q] with ||y − x||1 minimum. Suppose, on the contrary, that p(u) = q(u) and x(u) > y(u) for some u ∈ V . By (M-EXC[Z]) there exists v ∈ supp− (x − y) such that f (x) + f (y) ≥ f (x − χu + χv ) + f (y + χu − χv ).

(6.61)

By x ∈ arg min f [p] and y ∈ arg min f [q] we have f [p](x) ≤ f [p](x − χu + χv ),

f [q](y) ≤ f [q](y + χu − χv ),

(6.62)

and hence f (x − χu + χv ) ≥ f (x) + p(u) − p(v),

f (y + χu − χv ) ≥ f (y) − q(u) + q(v). (6.63)

Adding (6.61) and (6.63) yields f (x) + f (y) ≥ f (x) + f (y) + [p(u) − q(u)] + [q(v) − p(v)] ≥ f (x) + f (y). This shows that (6.61), (6.62), and (6.63) are satisfied in equalities. In particular, we have y + χu − χv ∈ arg min f [q], a contradiction to our choice of y. For a function f let f˜ be the function given by (6.4). It is easy to see that f˜ satisfies (M-GS[Z]) if and only if f satisfies the following:

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154

Chapter 6. M-Convex Functions (M -GS[Z]) If x ∈ arg min f [p − p01], p ≤ q, p0 ≤ q0 , and arg min f [q − q0 1] = ∅, there exists y ∈ arg min f [q − q0 1] such that (i) y(v) ≥ x(v) for every v ∈ V with p(v) = q(v), and (ii) y(V ) ≤ x(V ) if p0 = q0 ,

where p, q ∈ RV and p0 , q0 ∈ R. Note that (M -GS[Z]) is equivalent to the pair of (6.64) and (6.65) below: x ∈ arg min f [p], p ∈ RV , u ∈ V, α > 0, arg min f [p + αχu ] = ∅ =⇒ ∃ y ∈ arg min f [p + αχu ] : y(v) ≥ x(v)(∀ v ∈ V \ {u}), y(V ) ≤ x(V ), (6.64) x ∈ arg min f [p], p ∈ RV , α > 0, arg min f [p − α1] = ∅ =⇒ ∃ y ∈ arg min f [p − α1] : y(v) ≥ x(v) (∀ v ∈ V ).

(6.65)

Proposition 6.33. An M -convex function f ∈ M [Z → R] satisfies (M -GS[Z]). Proof. This follows from Proposition 6.32 applied to f˜ in (6.4). Note that (M -GS[Z]) ⇒ (M-GS[Z]) as well as M[Z → R] ⊆ M [Z → R]. Hence Proposition 6.32 is contained in Proposition 6.33 as a special case. The properties (M-GS[Z]) and (M -GS[Z]) characterize M-convex and M convex functions, respectively. Theorem 6.34. Let f : ZV → R ∪ {+∞} be a function that is convex extensible 47 and has a bounded nonempty effective domain. (1) If dom f ⊆ {x ∈ ZV | x(V ) = r} for some r ∈ Z, f is M-convex ⇐⇒ f satisfies (M-GS[Z]). (2) f is M -convex ⇐⇒ f satisfies (M -GS[Z]). Proof. The implications ⇒ in (1) and (2) have been shown in Propositions 6.32 and 6.33. We give a proof of ⇐ for (1) by using Theorem 6.30. It suffices to show that B = arg min f is an M-convex set, since (M-GS[Z]) for f implies this for f [−p] for any p ∈ RV . Since B = B ∩ ZV by the convex extensibility of f (see Proposition 3.18), this is further reduced to showing that every edge of polyhedron B is parallel to χu − χv for some u, v ∈ V (see (4.43)). Let E be an edge of B. By B = arg min f we have E ∩ ZV = arg min f [p] for some p ∈ RV . For two distinct integer points x, y on E, neither supp+ (x − y) nor supp− (x − y) is empty by B ⊆ {z ∈ ZV | z(V ) = r}. By (6.60) with u ∈ supp+ (x − y) and sufficiently small α > 0 there exists y¯ ∈ arg min f [q] such that y¯(v) ≥ x(v) (∀ v = u), y ). Since α > 0 where q = p + αχu . Note that x = y¯ since f [q](x) > f [q](y) ≥ f [q](¯ 47 It

will be shown in Theorem 6.42 that M -convex functions are convex extensible.

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155

is sufficiently small, we have y¯ ∈ arg min f [p] from y¯ ∈ arg min f [q]. This means that y¯ ∈ E and that x− y¯ is a scalar multiple of x− y. In particular, supp+ (x− y) = supp+ (x − y¯) = {u}. Similarly, supp− (x − y) = {v} for some v. Since x(V ) = y(V ), this means that x − y is a scalar multiple of χu − χv . A function f : ZV → R ∪ {+∞} is said to have the stepwise gross substitutes property (SWGS) if it satisfies the following: (M -SWGS[Z]) For x ∈ arg min f [p], p ∈ RV , and u ∈ V , at least one of (i) or (ii) holds true: (i) x ∈ arg min f [p + αχu ] for any α ≥ 0, (ii) there exist α ≥ 0 and y ∈ arg min f [p + αχu ] such that y(u) = x(u) − 1 and y(v) ≥ x(v) for all v ∈ V \ {u}. This property also characterizes M -convex functions. Proposition 6.35. An M -convex function f ∈ M [Z → R] satisfies (M -SWGS[Z]). Proof. We may assume p = 0. Suppose that (i) in (M -SWGS[Z]) fails, and let α∗ be the maximum value of α such that x ∈ arg min f [αχu ]. By the M-optimality criterion (Theorem 6.26 (2)), x ∈ arg min f [αχu ] if and only if f [αχu ](x) ≤ f [αχu ](x − χs + χt )

(∀ s, t ∈ V ∪ {0}),

which can be rewritten as α(χu (s) − χu (t)) ≤ Δf (x; t, s)

(∀ s, t ∈ V ∪ {0}).

Noting also that Δf (x; t, s) ≥ 0 (∀ s, t ∈ V ∪ {0}), we see α∗ =

min

t∈(V ∪{0})\{u}

Δf (x; t, u).

Let w ∈ (V ∪ {0}) \ {u} be such that α∗ = Δf (x; w, u) and put y = x − χu + χw . Then f [α∗ χu ](x) = f [α∗ χu ](y) as well as x ∈ arg min f [α∗ χu ]. Hence follows (ii) in (M -SWGS[Z]). Theorem 6.36. For a convex-extensible function f : ZV → R ∪ {+∞} with a nonempty effective domain, f is M -convex ⇐⇒ f satisfies (M -SWGS[Z]). Proof. The implication ⇒ was shown in Proposition 6.35. We give a proof of ⇐ by using Theorem 6.30 (2). It suffices to show that B = arg min f is an M convex set, since (M -SWGS[Z]) for f implies this for f [−p] for any p ∈ RV . Since B = B ∩ ZV by the convex extensibility of f (see Proposition 3.18), this is further reduced to showing that every edge of polyhedron B is parallel to χu − χv or χu

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for some u, v ∈ V (see (4.43)). Let E be an edge of B. By B = arg min f we have E ∩ ZV = arg min f [p] for some p ∈ RV . Let x and y be two distinct integer points on E with supp+ (x − y) = ∅. By (M -SWGS[Z]) with u ∈ supp+ (x − y), there ˆ(u) = x(u) − 1 and x ˆ(w) ≥ x(w) exist α ≥ 0 and x ˆ ∈ arg min f [p + αχu ] such that x (∀ w = u), since (i) of (M -SWGS[Z]) fails by f [p + χu ](y) < f [p + χu ](x). Since f [p](ˆ x) + α(x(u) − 1) = f [p + αχu ](ˆ x) ≤ f [p + αχu ](y) ≤ f [p](y) + α(x(u) − 1), we have x ˆ ∈ arg min f [p]. This means that x ˆ ∈ E and that x − x ˆ is a scalar multiple ˆ) = {u}. If supp− (x − y) = ∅, of x − y. In particular supp+ (x − y) = supp+ (x − x then x − y is a scalar multiple of χu . Otherwise, a similar argument shows that supp− (x−y) = {v} for some v and there exists yˆ ∈ E ∩ZV such that yˆ(v) = y(v)−1 and yˆ(w) ≥ y(w) (∀ w = v). Since x − x ˆ is a scalar multiple of y − yˆ, we have x ˆ(v) = x(v) + β and yˆ(u) = y(u) + 1/β for some β > 0. We must have β = 1 since x ˆ(v) and yˆ(u) are integers. Therefore, x − y is a scalar multiple of χu − χv .

6.9

Proximity Theorem

Suppose that we have an optimization problem to solve and another optimization problem approximating the original problem. Proximity theorem is a generic term for a theorem that guarantees the existence of an optimal solution to the original problem in some neighborhood of an optimal solution to the approximate problem. Our optimization problem here is the minimization of an M-convex function f , and the approximation to it is the problem of (locally) minimizing the scaling of f with a positive integer α, denoted as f α in (6.47). Recall from Note 6.18 that the scaling of an M-convex function is not necessarily M-convex, and hence a local optimum of f α may not be a global optimum of f α . The following proximity theorem, named the M-proximity theorem, shows that a global optimum of the original function f exists in a neighborhood of a local optimum of f α . Theorem 6.37 (M-proximity theorem). Assume α ∈ Z++ and n = |V |. (1) Let f : ZV → R ∪ {+∞} be an M-convex function. If xα ∈ dom f satisfies f (xα ) ≤ f (xα + α(χv − χu ))

(∀ u, v ∈ V ),

(6.66)

then arg min f = ∅ and there exists x∗ ∈ arg min f with ||xα − x∗ ||∞ ≤ (n − 1)(α − 1).

(6.67)

(2) Let f : ZV → R∪{+∞} be an M -convex function. If xα ∈ dom f satisfies f (xα ) ≤ f (xα + α(χv − χu ))

(∀ u, v ∈ V ∪ {0}),

(6.68)

then arg min f = ∅ and there exists x∗ ∈ arg min f with ||xα − x∗ ||∞ ≤ n(α − 1).

(6.69)

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157

Proof. It suffices to prove (1) by showing that, for any γ ∈ R with γ > inf f , there exists some x∗ ∈ dom f satisfying f (x∗ ) ≤ γ and (6.67). Suppose that x∗ ∈ dom f minimizes ||x∗ − xα ||1 among all vectors satisfying f (x∗ ) ≤ γ. In the following, we fix v ∈ V and prove xα (v) − x∗ (v) ≤ (n − 1)(α − 1). (The inequality x∗ (v) − xα (v) ≤ (n − 1)(α − 1) can be shown similarly.) We may assume xα (v) > x∗ (v); put k = xα (v) − x∗ (v). Claim 1: There exist w1 , w2 , . . . , wk ∈ V \{v} and y0 (= xα ), y1 , . . . , yk ∈ dom f such that yi = yi−1 − χv + χwi ,

f (yi ) < f (yi−1 ) (i = 1, . . . , k).

(Proof of Claim 1) We prove the claim by induction on i. Suppose yi−1 ∈ dom f . By (M-EXC[Z]) for yi−1 , x∗ , and v ∈ supp+ (yi−1 − x∗ ), there exists wi ∈ supp− (yi−1 − x∗ ) ⊆ supp− (xα − x∗ ) ⊆ V \ {v} such that f (x∗ ) + f (yi−1 ) ≥ f (x∗ − χwi + χv ) + f (yi−1 + χwi − χv ). By the choice of x∗ we have f (x∗ + χv − χwi ) > f (x∗ ), and hence f (yi ) = f (yi−1 − χv + χwi ) < f (yi−1 ). Claim 2: For any w ∈ V \{v} with yk (w) > xα (w) and μ ∈ [0, yk (w)−xα (w)− 1]Z , we have f (xα − (μ + 1)(χv − χw )) < f (xα − μ(χv − χw )). (Proof of Claim 2) We prove this by induction on μ. For μ ∈ [0, yk (w)−xα (w)− 1]Z , put x = xα − μ(χv − χw ) and assume x ∈ dom f . Let j∗ (1 ≤ j∗ ≤ k) be the largest index such that wj∗ = w. Then yj∗ (w) = yk (w) > x (w) and supp− (yj∗ − x ) = {v}. (M-EXC[Z]) implies f (x ) + f (yj∗ ) ≥ f (x − χv + χw ) + f (yj∗ + χv − χw ). By Claim 1 we have f (yj∗ + χv − χw ) > f (yj∗ ). This establishes Claim 2. Claim 2 and (6.66) imply f (xα − μw (χv − χw )) < · · · < f (xα − (χv − χw )) < f (xα ) ≤ f (xα − α(χv − χw )) for any w with μw ≡ yk (w) − xα (w) > 0. Hence yk (w) − xα (w) ≤ α − 1 for all w ∈ V \ {v}. Then we obtain  (yk (w) − xα (w)) ≤ (n − 1)(α − 1), xα (v) − x∗ (v) = xα (v) − yk (v) = w∈V \{v}

where the second equality is by xα (V ) = yk (V ). Example 6.38. The M-proximity theorem is illustrated for the univariate M convex function in Fig. 6.1 in section 6.4, where α = 2. Obviously, xα = 0 is the minimizer of f α satisfying (6.68) and x∗ = 1 is the minimizer of f . We have |xα − x∗ | = 1 = n(α − 1), in agreement with (6.69). The minimizer cut theorem (Theorem 6.28) can be adapted to scaling. Theorem 6.28, except for (3), is a special case of the following theorem with α = 1.

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Theorem 6.39 (M-minimizer cut with scaling). Let f : ZV → R ∪ {+∞} be an M-convex function with arg min f = ∅, and assume α ∈ Z++ and n = |V |. (1) For x ∈ dom f and v ∈ V , let u ∈ V be such that f (x + α(χv − χu )) = min f (x + α(χv − χs )). s∈V

Then there exists x∗ ∈ arg min f with x∗ (u) ≤ x(u) − α(1 − χv (u)) + (n − 1)(α − 1). (2) For x ∈ dom f and u ∈ V , let v ∈ V be such that f (x + α(χv − χu )) = min f (x + α(χt − χu )). t∈V

Then there exists x∗ ∈ arg min f with x∗ (v) ≥ x(v) + α(1 − χu (v)) − (n − 1)(α − 1). Proof. We prove (2), while (1) can be proved similarly. Put xα = x + α(χv − χu ). We may assume max{x∗ (v) | x∗ ∈ arg min f } < xα (v); otherwise we are done. Let x∗ be an element of arg min f with x∗ (v) maximum and k = xα (v) − x∗ (v) (≥ 1). The rest of the proof is the same as the proof of Theorem 6.37 (from Claim 1 until the end). The algorithmic use of the above theorems, M-proximity and minimizer cut with scaling, is shown in sections 10.1.2 and 10.1.4, respectively. Note 6.40. An 1 -norm version of Theorem 6.37 (1), with (6.67) replaced with ||xα − x∗ ||1 ≤

n2 (α − 1), 2

(6.70)

can be obtained from a slight modification of the proof; see Murota–Tamura [162].

Note 6.41. The M-proximity theorem (Theorem 6.37) is closely related to the result of Hochbaum [90]. See also Moriguchi–Shioura [134].

6.10

Convex Extension

This section establishes one of the major properties of M-convex functions—that they can be extended to convex functions in real variables. The extensibility to convex functions is by no means obvious from the definition of M-convex functions; note that the exchange axiom (M-EXC[Z]) refers only to function values on integer points. The convex extension of an M-convex function can be obtained by piecing together M-convex polyhedra in a consistent way.

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The first theorem shows that the convex extension of an M-convex function can be constructed locally. Theorem 6.42. An M -convex function is integrally convex. In particular, an M -convex function is convex extensible. Proof. It suffices to consider an M-convex function f . The restriction of f to any bounded integer interval [a, b], denoted by f[a,b] , is an M-convex function (Proposition 6.14). For any p ∈ RV , arg min(f[a,b] [−p]) is an M-convex set by Proposition 6.29, and hence it is an integrally convex set by Theorem 4.24. Therefore, f[a,b] is an integrally convex function by Theorem 3.29. This implies the integral convexity of f by Proposition 3.19. The next theorem characterizes the convex extension of an M-convex function as a collection of M-convex polyhedra. Theorem 6.43. Let f : ZV → R ∪ {+∞} be a function with dom f = ∅ and f be its convex closure. (1) ⎧ ⎨ (i) f is convex extensible (3.57), f is M-convex ⇐⇒ (ii) for every p ∈ RV , arg min f [−p] is ⎩ an M-convex polyhedron if it is not empty. ⎧ ⎨ (i) f is convex extensible (3.57), f is M -convex ⇐⇒ (ii) for every p ∈ RV , arg min f [−p] is ⎩ an M -convex polyhedron if it is not empty. (2)

Proof. It suffices to prove (1). The implication ⇒ is due to Theorem 6.42 and Proposition 6.29. The converse ⇐ can be established by Theorem 6.30 applied to the restriction of f to every bounded integer interval. By integral convexity, the convex extension f (x) of an M-convex function f can be represented as a convex combination of f (y) with y ∈ N (x), where N (x) is the integral neighborhood of x ∈ RV defined in (3.58). The following theorem states that we can use a single set of convex combination coefficients for a pair of M -convex functions. This fact, though technical, is crucial in establishing the separation theorem for M -convex functions (Theorem 8.15). Theorem 6.44. For two M -convex functions f1 , f2 ∈ M [Z → R] and x ∈ RV , there exists λ = (λy | y ∈ N (x)) such that   λy y = x, λy = 1, λy ≥ 0 (y ∈ N (x)), (6.71) y∈N (x)

fi (x) = f˜i (x) =

y∈N (x)



y∈N (x)

λy fi (y)

(i = 1, 2).

(6.72)

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Proof. We may assume f1 and f2 to be M-convex and x ∈ dom f1 ∩ dom f2 . For i = 1, 2, let (pi , αi ) ∈ RV × R be such that pi , y + αi ≤ fi (y) (y ∈ N (x)),

pi , x + αi = f˜i (x)

(see (3.61)). Then Bi = {y ∈ N (x) | pi , y + αi = fi (y)} = N (x) ∩ arg min fi [−pi ] is an M-convex set. Since x ∈ B1 ∩ B2 , where B1 ∩ B2 = B1 ∩ B2 by Theorem 4.22, there exists λ = (λy | y ∈ N (x)) satisfying {y | λy > 0} ⊆ B1 ∩ B2 and (6.71). Such a λ also satisfies (6.72) by the complementarity (Theorem 3.10 (3)), as in the proof of Theorem 3.29.

6.11

Polyhedral M-Convex Functions

As we have seen, M-convex functions on the integer lattice can be extended to convex functions in real variables. The convex extension of an M-convex function is a polyhedral convex function when restricted to a finite interval. Motivated by this we define here the concept of M-convexity for polyhedral convex functions in general and show that major properties of M-convex functions survive in this generalization. A polyhedral convex function f : RV → R ∪ {+∞} with domR f = ∅ is said to be M-convex if it satisfies the following exchange property: (M-EXC[R]) For x, y ∈ domR f and u ∈ supp+ (x − y), there exist v ∈ supp− (x − y) and a positive number α0 ∈ R++ such that f (x) + f (y) ≥ f (x − α(χu − χv )) + f (y + α(χu − χv )) for all α ∈ [0, α0 ]R . Note that, if the inequality above holds for α = α0 , it holds for all α ∈ [0, α0 ]R by convexity of f . With the notation f  (z; v, u) = f  (z; χv − χu )

(z ∈ dom f ; u, v ∈ V )

(6.73)

for directional derivatives (see (3.24)), (M-EXC[R]) can be rewritten as follows: (M-EXC [R]) For x, y ∈ domR f , max

min

[f  (x; v, u) + f  (y; u, v)] ≤ 0.

u∈supp+ (x−y) v∈supp− (x−y)

(6.74)

We denote by M[R → R] the set of polyhedral M-convex functions. Polyhedral M-concave functions are defined in an obvious way. An M-convex function on integer points naturally induces a polyhedral Mconvex function via convex extension (which exists by Theorem 6.42).

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161

Theorem 6.45. The convex extension f of an M-convex function f ∈ M[Z → R] on the integer lattice is a polyhedral M-convex function, i.e., f ∈ M[R → R], provided that f is polyhedral. Proof. The proof is given later in Note 8.8. Example 6.46. The convex extension f of an M-convex function f ∈ M[Z → R] may consist of an infinite number of linear pieces, in which case f is not polyhedral convex. For example, we have f ∈ M[Z → R] and f ∈ / M[R → R] for f : Z2 → R ∪ {+∞} defined by  2 x(1) (x(1) + x(2) = 0), f (x) = +∞ (otherwise). It is worth noting that, if domZ f is bounded, f is polyhedral and therefore f ∈ M[R → R] by Theorem 6.45. We now define integrality for polyhedral convex functions in general. By an integral polyhedral convex function we mean a polyhedral convex function f such that arg min f [−p] is an integral polyhedron for every p ∈ RV .

(6.75)

We say that a polyhedral convex function f has dual integrality, or is a dual-integral polyhedral convex function, if its conjugate function f • has integrality (6.75). Since arg min f • [−x] = ∂R f (x), as in (3.30), f has dual integrality if and only if ∂R f (x) is an integral polyhedron for every x ∈ domR f .

(6.76)

We denote by C[Z|R → R] and C[R → R|Z] the sets of univariate polyhedral convex functions with integrality (6.75) and dual integrality (6.76), respectively. Polyhedral M-convex functions with integrality (6.75) are referred to as integral polyhedral M-convex functions, the set of which is denoted by M[Z|R → R]. Polyhedral M-convex functions with dual integrality (6.76) are referred to as dualintegral polyhedral M-convex functions, the set of which is denoted by M[R → R|Z]. By Theorems 6.45 and 6.43, an integral polyhedral M-convex function is nothing but a polyhedral M-convex function that can be obtained as the convex extension of an M-convex function on integer points. Therefore, we have M[Z|R → R] ⊆ M[R → R],

M[Z|R → R] → M[Z → R],

(6.77)

where the second expression means that there exists an injection from M[Z|R → R] to M[Z → R], representing an embedding of M[Z|R → R] into M[Z → R]. The effective domain of a polyhedral M-convex function is an M-convex polyhedron lying on a hyperplane {x ∈ RV | x(V ) = r} for some r ∈ R. Hence, polyhedral M -convex functions can be defined as the projection of polyhedral Mconvex functions, just as M -convex functions on integer points are defined from

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M-convex functions via (6.4). We denote by M [R → R] the set of polyhedral M -convex functions and by M [Z|R → R] the set of integral polyhedral M -convex functions. The relationship between M and M is described by Mn ⊂ Mn  Mn+1 , where Mn and Mn denote, respectively, the sets of polyhedral M-convex functions and polyhedral M -convex functions in n variables. The following are the R-counterparts of (M -EXC[Z]) and (M -EXC [Z]): (M -EXC[R]) For x, y ∈ domR f and u ∈ supp+ (x − y), there exist v ∈ supp− (x − y) ∪ {0} and a positive number α0 ∈ R++ such that f (x) + f (y) ≥ f (x − α(χu − χv )) + f (y + α(χu − χv )) for all α ∈ [0, α0 ]R , where χ0 = 0. (M -EXC [R]) For x, y ∈ domR f , max

min

[f  (x; v, u) + f  (y; u, v)] ≤ 0,

u∈supp+ (x−y) v∈supp− (x−y)∪{0}

(6.78)

where f  (x; 0, u) = f  (x; −χu ),

f  (y; u, 0) = f  (y; χu ).

Theorem 6.47. For a polyhedral convex function f : RV → R ∪ {+∞} with domR f = ∅, we have polyhedral M -convexity ⇐⇒ (M -EXC[R]) ⇐⇒ (M -EXC [R]). Theorem 6.48. A polyhedral M-convex function is polyhedral M -convex. Conversely, a polyhedral M -convex function is polyhedral M-convex if and only if the effective domain is contained in {x ∈ RV | x(V ) = r} for some r ∈ R. Almost all properties of M-convex functions on integer points carry over to polyhedral M-convex functions. To be specific, Theorems 6.13, 6.15, 6.19, and 6.26 and Proposition 6.29 are adapted as follows. Note, however, that the proofs are not straightforward adaptations; see Murota–Shioura [152]. For a subset U ⊆ V , the restriction fU : RU → R ∪ {+∞}, the projection U f : RU → R ∪ {±∞}, and the aggregation f U∗ : RU × R → R ∪ {±∞} are defined similarly to (6.40), (6.41), and (6.42). Note in Theorem 6.49 (2) below that a scaling factor β is allowed, unlike in the discrete case (cf. Note 6.18). Theorem 6.49. Let f, f1 , f2 ∈ M[R → R] be polyhedral M-convex functions. (1) For λ ∈ R++ , λf is polyhedral M-convex. (2) For a ∈ RV and β ∈ R \ {0}, f (a + βx) is polyhedral M-convex in x. (3) For p ∈ RV , f [−p] is polyhedral M-convex.

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6.11. Polyhedral M-Convex Functions (4) For ϕv ∈ C[R → R] (v ∈ V ),  f˜(x) = f (x) + ϕv (x(v))

163

(x ∈ RV )

(6.79)

v∈V

is polyhedral M-convex provided domR f˜ = ∅. (5) For a, b ∈ (R ∪ {±∞})V , the restriction f[a,b] to the real interval [a, b] is polyhedral M-convex provided domR f[a,b] = ∅. (6) For U ⊆ V , the restriction fU is polyhedral M-convex provided domR fU = ∅. (7) For U ⊆ V , the aggregation f U∗ is polyhedral M-convex provided U∗ f > −∞. (8) The infimal convolution f˜ = f1 2 f2 is polyhedral M-convex provided ˜ f > −∞. Theorem 6.50. Let f, f1 , f2 ∈ M [R → R] be polyhedral M -convex functions. (1) Operations (1)–(8) of Theorem 6.49 are valid for polyhedral M -convex functions. (2) For U ⊆ V , the projection f U is polyhedral M -convex provided f U > −∞. Theorem 6.51. A polyhedral M -convex function f ∈ M [R → R] is supermodular; i.e., f (x) + f (y) ≤ f (x ∨ y) + f (x ∧ y) (x, y ∈ RV ). Theorem 6.52 (M-optimality criterion). (1) For a polyhedral M-convex function f ∈ M[R → R] and x ∈ domR f , we have f (x) ≤ f (y) (∀ y ∈ RV ) ⇐⇒ f  (x; −χu + χv ) ≥ 0 (∀ u, v ∈ V ). (2) For a polyhedral M -convex function f ∈ M [R → R] and x ∈ domR f , we have   f (x; −χu + χv ) ≥ 0 (∀ u, v ∈ V ), V f (x) ≤ f (y) (∀ y ∈ R ) ⇐⇒ f  (x; ±χv ) ≥ 0 (∀ v ∈ V ). Proposition 6.53. Let f ∈ M[R → R] be a polyhedral M-convex function. For any p ∈ RV , arg min f [−p] is an M-convex polyhedron if it is not empty. The property in Proposition 6.53 characterizes polyhedral M-convexity, to be shown in Theorem 6.63. Note 6.54. Here are two remarks on α0 in (M-EXC[R]). First, for an integral polyhedral M-convex function f ∈ M[Z|R → R], we can take α0 = 1. Second, if (M-EXC[R]) is true at all, we can take α0 = 12 (x(u) − y(u))/|supp− (x − y)| independently of f ; see Murota–Shioura [152] for the proof.

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Note 6.55. The proviso f U∗ > −∞ in Theorem 6.49 (7) can be weakened to f U∗ (x0 ) > −∞ for some x0 . The same can be said for f˜ > −∞ in Theorem 6.49 (8) and f U > −∞ in Theorem 6.50 (2).

6.12

Positively Homogeneous M-Convex Functions

There exists a one-to-one correspondence between positively homogeneous M-convex functions and distance functions satisfying the triangle inequality. We denote by 0 M[R → R] the set of polyhedral M-convex functions that are positively homogeneous in the sense of (3.32) and by 0 M[Z|R → R] the set of integral polyhedral M-convex functions that are positively homogeneous. Also we denote by 0 M[Z → R] the set of M-convex functions f ∈ M[Z → R] on integer points such that the convex extensions f are positively homogeneous. These three families of functions can be identified with each other, i.e., 0 M[Z

→ R]  0 M[Z|R → R] = 0 M[R → R],

(6.80)

by the following proposition. We introduce yet another notation, 0 M[Z → Z], for the set of integer-valued functions belonging to 0 M[Z → R]. Proposition 6.56. (1) 0 M[Z|R → R] = 0 M[R → R]. (2) The convex extension of a function in 0 M[Z → R] belongs to 0 M[R → R]. Proof. (1) Take f ∈ 0 M[R → R]. For any p ∈ RV , arg min f [−p] is a cone that is an M-convex polyhedron (or empty) by Proposition 6.53. Hence, arg min f [−p] = B(ρ) for a {0, +∞}-valued submodular set function ρ; see section 4.8. This shows the integrality of arg min f [−p], and therefore f ∈ 0 M[Z|R → R]. (2) Take f ∈ 0 M[Z → R]. Since f is integrally convex and f is positively homogeneous, f can be represented as the maximum of a finite number of linear functions. Hence, f is polyhedral and f ∈ M[R → R] by Theorem 6.45. A positively homogeneous M-convex function f induces a distance function γ = γf satisfying the triangle inequality by γf (u, v) = f (χv − χu )

(u, v ∈ V ).

(6.81)

More precisely, we have the following, where T [R] and T [Z] denote respectively the sets of real-valued and integer-valued distance functions with the triangle inequality. Proposition 6.57. (1) For f ∈ 0 M[R → R], we have γf ∈ T [R]. (2) For f ∈ 0 M[Z → Z], we have γf ∈ T [Z]. Proof. For (1) we apply (M-EXC[R]) to x = χv3 − χv2 , y = χv2 − χv1 , and u = v1 , where we can take α = 1 by Proposition 6.56 (1) and Note 6.54. This yields the triangle inequality (5.2). For (2) we use (M-EXC[Z]) in a similar manner.

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165

Conversely, a distance function satisfying the triangle inequality induces a positively homogeneous M-convex function. For γ ∈ T [R], we define γˆ : RV → R ∪ {+∞} by ⎧ ) ⎫ )  ⎨ ⎬ ) γˆ (x) = inf λuv γ(u, v))) λuv (χv − χu ) = x, λuv ∈ R+ (u, v ∈ V ) , λ ⎩ ⎭ ) u,v∈V u,v∈V (6.82) which is called the extension of γ. Proposition 5.1 as well as its proof shows γˆ(χv − χu ) = γ(u, v)

(u, v ∈ V ),

γˆ(x) = sup{ p, x | p ∈ D(γ)}

(6.83)

(x ∈ R ), V

(6.84)

where D(γ) is the L-convex polyhedron (5.4) associated with γ. Denote by γˆZ : ZV → R ∪ {+∞} the restriction of γˆ to ZV , and note that for x ∈ ZV we may assume λuv ∈ Z+ in (6.82), as is explained in the proof of Proposition 5.1. Proposition 6.58. (1) For γ ∈ T [R], we have γˆ ∈ 0 M[R → R]. (2) For γ ∈ T [Z], we have γˆZ ∈ 0 M[Z → Z]. Proof. Expression (6.82) is a special case of the M-convex function (2.37) appearing in network flow problems (section 2.2), where T = V , A = {a = (u, v) | u, v ∈ V ; u = v}, and fuv (ξ) = γ(u, v)ξ (for ξ ≥ 0) and +∞ (for ξ < 0). See also Note 2.19. The next theorem shows a one-to-one correspondence between positively homogeneous M-convex functions and distance functions satisfying the triangle inequality. Theorem 6.59. For 0 M = 0 M[R → R] and T = T [R], the mappings Φ : 0 M → T and Ψ : T → 0 M defined by Φ : f → γf in (6.81),

Ψ : γ → γˆ in (6.82)

are inverse to each other, establishing a one-to-one correspondence between 0 M and T . The same statement is true for 0 M = 0 M[Z → Z] and T = T [Z]. Proof. For γ ∈ T , we have Ψ(γ) ∈ 0 M by Proposition 6.58 and Φ ◦ Ψ(γ) = γ by (6.83). For f ∈ 0 M, we have Φ(f ) ∈ T by Proposition 6.57. Since f is a positively homogeneous convex function, we have ⎞ ⎛   λuv (χv − χu )⎠ ≤ λuv f (χv − χu ) f (x) = f ⎝ u,v∈V

u,v∈V

whenever u,v∈V λuv (χv − χu ) = x and λuv ≥ 0 (u, v ∈ V ). This implies f ≤ Ψ ◦ Φ(f ). In the case of 0 M = 0 M[Z → Z] and T = T [Z], the opposite inequality

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f ≥ Ψ◦Φ(f ) is given by Proposition 6.25, whereas, in the case of 0 M = 0 M[R → R] and T = T [R], the inequality can be established by an argument similar to the proof of Proposition 6.23.

6.13

Directional Derivatives and Subgradients

Directional derivatives and subgradients of M-convex functions are considered in this section. For a polyhedral M-convex function f , the directional derivative f  (x; d) is a positively homogeneous M-convex function in d, and the subgradients of f at a point form an L-convex polyhedron. Furthermore, each of these properties characterizes M-convexity. We start with directional derivatives of a polyhedral M-convex function f ∈ M[R → R]. Recall from (3.25) that, for each x ∈ domR f , there exists ε > 0 such that (||d||1 ≤ ε). (6.85) f (x + d) − f (x) = f  (x; d) Proposition 6.60. For f ∈ M[R → R] and x ∈ domR f , we have f  (x; ·) ∈ 0 M[R → R]. Proof. By (6.85), f  (x; ·) has the exchange property in the neighborhood of d = 0. Then the claim follows from the positive homogeneity of f  (x; ·). For a function f : ZV → R ∪ {+∞} and a point x ∈ domZ f , we define ∂R f (x) = {p ∈ RV | f (y) − f (x) ≥ p, y − x (∀ y ∈ ZV )}

(6.86)

and call it the subdifferential of f at x (cf. (3.23)). An element of ∂R f (x) is called a subgradient of f at x. If f is convex extensible, we have ∂R f (x) = ∂R f (x)

(x ∈ domZ f ),

(6.87)

where f is the convex extension of f . The set of integer-valued subgradients ∂Z f (x) = ∂R f (x) ∩ ZV

(6.88)

is called the integer subdifferential of f at x ∈ domZ f . Directional derivatives and subdifferentials of M-convex functions are given as follows. It is recalled that L0 [R], L0 [Z|R], L0 [Z], and M[R → R|Z] denote, respectively, the sets of L-convex polyhedra, integral L-convex polyhedra, L-convex sets, and dual-integral polyhedral M-convex functions. Also recall the definition of γˆ in (6.82). Theorem 6.61. (1) For f ∈ M[R → R] and x ∈ domR f , define γf,x (u, v) = f  (x; −χu + χv ) (u, v ∈ V ). Then γf,x ∈ T [R],

∂R f (x) = D(γf,x ) ∈ L0 [R],

f  (x; ·) = γˆf,x (·),

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167

and ∂R f (x) = ∅ in particular. If f ∈ M[R → R|Z], then γf,x ∈ T [Z],

∂R f (x) ∈ L0 [Z|R].

(2) For f ∈ M[Z → R] and x ∈ domZ f , define γf,x (u, v) = f (x − χu + χv ) − f (x) (u, v ∈ V ). Then γf,x ∈ T [R],



∂R f (x) = D(γf,x ) ∈ L0 [R],

f (x; ·) = γˆf,x (·),

and ∂R f (x) = ∅ in particular. If f ∈ M[Z → Z], then γf,x ∈ T [Z],

∂R f (x) ∈ L0 [Z|R],

∂Z f (x) ∈ L0 [Z],

∂R f (x) = ∂Z f (x),

and ∂Z f (x) = ∅ in particular. Proof. (1) Proposition 6.60 shows f  (x; ·) ∈ 0 M[R → R], from which follows γf,x ∈ T [R] by Proposition 6.57. By the definition of a subdifferential and Theorem 6.52 (M-optimality criterion) we see p ∈ ∂R f (x) ⇐⇒ f (x + d) − f (x) ≥ p, d

(∀ d ∈ RV )

⇐⇒ f  (x; −χu + χv ) ≥ p, −χu + χv (∀ u, v ∈ V ) ⇐⇒ p(v) − p(u) ≤ γf,x (u, v) (∀ u, v ∈ V ) ⇐⇒ p ∈ D(γf,x ). We have D(γf,x ) ∈ L0 [R] by (5.18) and f  (x; ·) = γˆf,x (·) by (3.31), (3.33), and (6.84). If f ∈ M[R → R|Z], ∂R f (x) is an integral polyhedron by (6.76) and γf,x (u, v) = sup{p(v) − p(u) | p ∈ ∂R f (x)} ∈ Z. (2) Applying (M-EXC[Z]) to x + χv3 − χv2 , x + χv2 − χv1 , and u = v1 shows the triangle inequality (5.2), and hence γf,x ∈ T [R]. The rest of the proof is similar to (1), where we use Theorem 6.26 (1) instead of Theorem 6.52 and Theorem 5.5 instead of (5.18). The following fact shows the consistency of (1) and (2) in Theorem 6.61. Proposition 6.62. For f ∈ M[Z|R → R] and x ∈ domR f ∩ ZV , we have f  (x; −χu + χv ) = f (x − χu + χv ) − f (x) for u, v ∈ V . Proof. This follows from integrality (6.75) and (4.40). The following theorem affords characterizations of polyhedral M-convex functions in terms of the M-convexity of directional derivatives, the L-convexity of subdifferentials, and the M-convexity of minimizers. Theorem 6.63. For a polyhedral convex function f : RV → R ∪ {+∞} with domR f = ∅, the four conditions (a), (b), (c), and (d) below are equivalent. (a) f ∈ M[R → R].

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Y = f (x) x Figure 6.3. Quasi-convex function.

(b) f  (x; ·) ∈ 0 M[R → R] for every x ∈ domR f . (c) ∂R f (x) ∈ L0 [R] for every x ∈ domR f . (d) arg min f [−p] ∈ M0 [R] for every p ∈ RV with inf f [−p] > −∞. Proof. (a) ⇒ (b) is by Proposition 6.60, (a) ⇒ (c) by Theorem 6.61, and (a) ⇒ (d) by Proposition 6.53. The rest is proved later in Note 8.7. An integrality consideration in the equivalence of (a) and (d) in the above theorem yields a characterization of integral polyhedral M-convex functions. Theorem 6.64. For a polyhedral convex function f : RV → R ∪ {+∞} with domR f = ∅, the two conditions (a) and (d) below are equivalent. (a) f ∈ M[Z|R → R]. (d) arg min f [−p] ∈ M0 [Z|R] for every p ∈ RV with inf f [−p] > −∞. Note 6.65. By Theorem 6.61 we can identify fˇ(x, y) in Proposition 6.25 as the directional derivative of the convex extension f of f ∈ M[Z → R]. That is, we  have fˇ(x, y) = f (x; y − x).

6.14

Quasi M-Convex Functions

Quasi M-convex functions are introduced as a generalization of M-convex functions. The optimality criterion and the proximity theorem survive in this generalization. A function f : Rn → R ∪ {+∞} is said to be quasi convex if it satisfies max{f (x), f (y)} ≥ f (λx + (1 − λ)y)

(6.89)

whenever x, y ∈ domR f and 0 < λ < 1 and semistrictly quasi convex if max{f (x), f (y)} > f (λx + (1 − λ)y)

(6.90)

whenever x, y ∈ domR f , f (x) = f (y), and 0 < λ < 1. See Fig. 6.3 for an illustration of a (semistrictly) quasi-convex function.

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Quasi convexity is ordinal convexity in the sense that the definition involves no addition of function values, but relies only on comparisons. In this connection note that, if f (x) is convex and ϕ is a nondecreasing function representing a nonlinear scaling, then ϕ(f (x)) is quasi convex. Quasi-convex functions enjoy the following nice properties: • A strict local minimum of a quasi-convex function is a strict global minimum. • A local minimum of a semistrictly quasi-convex function is a global minimum. • Level sets of quasi-convex functions are convex sets. Due to these properties, quasi convexity also plays an important role in continuous optimization (see, e.g., Avriel–Diewert–Schaible–Zang [5]). The concept of quasi M-convexity is defined for a function f : ZV → R∪{+∞} as follows. Recall the exchange axiom for M-convex functions: (M-EXC[Z]) For x, y ∈ dom f and u ∈ supp+ (x − y), there exists v ∈ supp− (x − y) satisfying Δf (x; v, u) + Δf (y; u, v) ≤ 0.

(6.91)

The sign patterns of Δf (x; v, u) and Δf (y; u, v) compatible with (implied by) inequality (6.91) are as follows: Δf (x; v, u) \ Δf (y; u, v) − 0 +

− # # #

0 # # ×

+ # × ×

Here # and × denote possible and impossible cases, respectively. Relaxing condition (6.91) to compatible sign patterns leads to two versions of quasi M-convex functions. We say that a function f : ZV → R ∪ {+∞} with dom f = ∅ is quasi M-convex if it satisfies the following: (QM) For x, y ∈ dom f and u ∈ supp+ (x − y), there exists v ∈ supp− (x − y) satisfying Δf (x; v, u) ≤ 0 or Δf (y; u, v) ≤ 0. Similarly, a function f : ZV → R ∪ {+∞} with dom f = ∅ is semistrictly quasi M-convex if it satisfies the following: (SSQM) For x, y ∈ dom f and u ∈ supp+ (x − y), there exists v ∈ supp− (x − y) satisfying Δf (x; v, u) < 0 or Δf (y; u, v) < 0

or Δf (x; v, u) = Δf (y; u, v) = 0.

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Example 6.66. A quasi M-convex function arises from a nonlinear scaling of an M-convex function. For an M-convex function f : ZV → R ∪ {+∞} and a function ϕ : R → R ∪ {+∞}, define f˜ : ZV → R ∪ {+∞} by  ϕ(f (x)) (x ∈ dom f ), ˜ f (x) = (6.92) +∞ (x ∈ / dom f ). Then f˜ satisfies (QM) if ϕ is nondecreasing and (SSQM) if ϕ is strictly increasing.

The following weaker variants of (QM) and (SSQM) turn out to be useful for our subsequent discussion: (QMw ) For distinct x, y ∈ dom f , there exist u ∈ supp+ (x − y) and v ∈ supp− (x − y) satisfying Δf (x; v, u) ≤ 0

or Δf (y; u, v) ≤ 0.

(SSQMw ) For distinct x, y ∈ dom f , there exist u ∈ supp+ (x − y) and v ∈ supp− (x − y) satisfying Δf (x; v, u) < 0

or Δf (y; u, v) < 0

or Δf (x; v, u) = Δf (y; u, v) = 0.

The property (QMw ) can be expressed in two alternative forms below. The first (6.93) may be regarded as a variant of (6.89) with discreteness in direction, and the second (6.94) is similar to (6.50) in section 6.6. Theorem 6.67. For f : ZV → R ∪ {+∞}, (QMw ) is equivalent to each of the following conditions: max{f (x), f (y)} ≥

min

min

{f (x − χu + χv ), f (y + χu − χv )}

u∈supp+ (x−y) v∈supp− (x−y)

(∀ x, y ∈ dom f with x = y), f (x) ≥

min

min

u∈supp+ (x−y) v∈supp− (x−y)

(6.93)

f (x − χu + χv )

(∀ x, y ∈ dom f with x = y, f (x) ≥ f (y)). (6.94) Proof. Obviously, (6.94) implies (QMw ) and (6.93). We prove (QMw ) =⇒ (6.94) and (6.93) =⇒ (6.94) by induction on ||x − y||1 . Suppose x, y ∈ dom f and f (x) ≥ f (y); we may assume ||x−y||1 > 2. If (QMw ) is true, there exist some u ∈ supp+ (x− y) and v ∈ supp− (x − y) such that Δf (x; v, u) ≤ 0 or Δf (y; u, v) ≤ 0; in the latter case the induction hypothesis for x and y  = y + χu − χv yields Δf (x; v  , u ) ≤ 0 for some u ∈ supp+ (x − y  ) ⊆ supp+ (x − y) and v  ∈ supp− (x − y  ) ⊆ supp− (x − y). If (6.93) is true, there exist u ∈ supp+ (x − y) and v ∈ supp− (x − y) such that Δf (x; v, u) ≤ 0 or f (y + χu − χv ) ≤ f (x); in the latter case we have f (x) ≥ f (y  ) for y  = y + χu − χv and the induction hypothesis yields Δf (x; v  , u ) ≤ 0 for some u ∈ supp+ (x − y  ) ⊆ supp+ (x − y) and v  ∈ supp− (x − y  ) ⊆ supp− (x − y).

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The relationship among various versions of quasi M-convex functions is summarized as follows. The second statement below shows that all the conditions are equivalent for f if they are imposed on every perturbation of f by a linear function. Theorem 6.68. For f : ZV → R ∪ {+∞} the following implications hold true. (1) (M-EXC[Z]) =⇒ (SSQM) =⇒ (QM) ! ⇓ ⇓ (M-EXCw [Z]) =⇒ (SSQMw ) =⇒ (QMw ) (2) f satisfies (M-EXC[Z]) ⇐⇒ ∀ p ∈ RV : f [p] satisfies (QMw ). Proof. (1) The equivalence of (M-EXC[Z]) and (M-EXCw [Z]) is due to Theorem 6.5. The remaining implications are obvious. (2) Combining Theorems 6.72 and 6.74 below establishes this. The quasi M-convexity of a set B ⊆ ZV can be defined as the quasi Mconvexity of the indicator function δB : ZV → {0, +∞}. The properties (QM) and (QMw ) for δB correspond respectively to the following properties of B: (Q-EXC) For x, y ∈ B and u ∈ supp+ (x−y), there exists v ∈ supp− (x− y) such that x − χu + χv ∈ B or y + χu − χv ∈ B. (Q-EXCw ) For distinct x, y ∈ B, there exist u ∈ supp+ (x − y) and v ∈ supp− (x − y) such that x − χu + χv ∈ B or y + χu − χv ∈ B. Proposition 6.69. A set B ⊆ ZV satisfies (Q-EXCw ) if and only if, for distinct x, y ∈ B, there exist u ∈ supp+ (x − y) and v ∈ supp− (x − y) with x − χu + χv ∈ B. Proof. Theorem 6.67 for f = δB reduces to this statement. Proposition 6.70. For a set B ⊆ ZV satisfying (Q-EXCw ), we have x(V ) = y(V ) for any x, y ∈ B. Proof. The proof is easy (similar to that of Proposition 4.1). Example 6.71. Whereas we have the obvious implications (B-EXC[Z]) ⇒ (QEXC) ⇒ (Q-EXCw ), these conditions are not equivalent. For instance, B1 = {χS | S = {1, 2}, {2, 3}, {3, 4}} satisfies (Q-EXC) and not (B-EXC[Z]), and B2 = {χS | S = {1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {4, 5, 6}} satisfies (Q-EXCw ) and not (Q-EXC).

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The weaker version (QMw ) of quasi M-convexity for functions can be characterized by the corresponding quasi M-convexity (Q-EXCw ) of level sets. For any α ∈ R ∪ {+∞}, the level set is defined as L(f, α) = {x ∈ ZV | f (x) ≤ α}.

(6.95)

Note that L(f, +∞) = dom f and L(f, α0 ) = arg min f for α0 = min f . Theorem 6.72. A function f : ZV → R ∪ {+∞} satisfies (QMw ) if and only if the level set L(f, α) satisfies (Q-EXCw ) for all α ∈ R. Proof. [“only if”]: Let x and y be distinct elements of L(f, α). By (QMw ), we have Δf (x; v, u) ≤ 0 or Δf (y; u, v) ≤ 0 for some u ∈ supp+ (x − y) and v ∈ supp− (x − y). Then, x − χu + χv ∈ L(f, α) or y + χu − χv ∈ L(f, α). [“if”]: For any distinct x, y ∈ dom f with f (x) ≥ f (y), we have x − χu + χv ∈ L(f, f (x)) for some u ∈ supp+ (x − y) and v ∈ supp− (x − y) by (Q-EXCw ) and Proposition 6.69. Hence f (x − χu + χv ) ≤ f (x).

Proposition 6.73. If f satisfies (QMw ), then dom f satisfies (Q-EXCw ). Proof. In the proof of “only if” of Theorem 6.72, replace L(f, α) with dom f . An M-convex function can be characterized by quasi M-convexity of level sets of perturbed functions. Theorem 6.74. A function f : ZV → R ∪ {+∞} satisfies (M-EXC[Z]) if and only if the level set L(f [p], α) satisfies (Q-EXCw ) for all p ∈ RV and α ∈ R. Proof. The “only if” part follows from Theorem 6.72. To prove the “if” part, we first observe that Theorem 6.4 can be strengthened to a statement that (MEXC[Z]) and (M-EXCloc [Z]) are equivalent if dom f satisfies (Q-EXCw ). (This can be shown by modifying the proof of Claim 2 in the proof of Theorem 6.4.) Note that (Q-EXCw ) holds for dom f by Theorem 6.72 and Proposition 6.73. To show (MEXCloc [Z]), take x, y ∈ dom f with ||x−y||1 = 4 and put y = x−χu1 −χu2 +χv1 +χv2 with u1 , u2 , v1 , v2 ∈ V and {u1 , u2 } ∩ {v1 , v2 } = ∅. In the following we assume u1 = u2 and v1 = v2 (the other cases can be treated similarly). Consider a bipartite graph G = (V + , V − ; E) with vertex bipartition V + = {u1 , u2 }, V − = {v1 , v2 } and arc set E = {(ui , vj ) | Δf (x; vj , ui ) < +∞ (i, j = 1, 2)}. Claim 1: G has a perfect matching (of size 2). (Proof of Claim 1) It suffices to show that every vertex has an edge incident to it. Take p ∈ RV such that p(u1 ) + p(u2 ) − p(v1 ) − p(v2 ) = f (y) − f (x) and p(vj ) > p(u1 ) − Δf (x; vj , u1 ) for j = 1, 2 (and p(v) = 0 for v ∈ V \ {u1 , u2 , v1 , v2 }). Then f [p](x) = f [p](y) < f [p](x − χu1 + χvj ) for j = 1, 2. By (Q-EXCw ) for L(f [p], f [p](x)) we have f [p](x − χu2 + χvj ) ≤ f [p](x) < +∞ for some j ∈ {1, 2}. Hence there is an edge incident to u2 , and similarly for other vertices. Claim 2: Inequality (6.11) is satisfied.

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(Proof of Claim 2) By Claim 1 we may assume {(u1 , v1 ), (u2 , v2 )} ⊆ E. We can take p ∈ RV such that f [p](x) = f [p](y) and f [p](x−χu1 +χv1 ) = f [p](x−χu2 +χv2 ) (and p(v) = 0 for v ∈ V \ {u1 , u2 , v1 , v2 }). If {(u1 , v2 ), (u2 , v1 )} ⊆ E, we can choose p satisfying an additional condition f [p](x − χu1 + χv2 ) = f [p](x − χu2 + χv1 ), and then (Q-EXCw ) for L(f [p], f [p](x)) yields (6.11). If (u1 , v2 ) ∈ E and (u2 , v1 ) ∈ / E, we can choose p satisfying an additional condition f [p](x) < f [p](x − χu1 + χv2 ), and then (Q-EXCw ) for L(f [p], f [p](x)) yields (6.11). The remaining cases are similar. Next we turn to the minimization of quasi M-convex functions. The following properties, respectively weaker than (SSQM) and (SSQMw ), turn out to be relevant. (SSQM = ) For x, y ∈ dom f with f (x) = f (y) and u ∈ supp+ (x − y), there exists v ∈ supp− (x − y) satisfying Δf (x; v, u) < 0 or Δf (y; u, v) < 0

or Δf (x; v, u) = Δf (y; u, v) = 0.

+ (SSQM = w ) For x, y ∈ dom f with f (x) = f (y), there exist u ∈ supp (x− − y) and v ∈ supp (x − y) satisfying

Δf (x; v, u) < 0 or Δf (y; u, v) < 0

or Δf (x; v, u) = Δf (y; u, v) = 0.

The property (SSQM = w ) can be expressed in two alternative forms below. The first (6.96) may be regarded as a variant of (6.90) with discreteness in direction, and the second (6.97) is identical to (6.50) in section 6.6. Theorem 6.75. For f : ZV → R ∪ {+∞}, (SSQM = w ) is equivalent to each of the following conditions: max{f (x), f (y)} >

f (x) >

{f (x − χu + χv ), f (y + χu − χv )}

min

min

min

(∀ x, y ∈ dom f with f (x) = f (y)), (6.96) min f (x − χu + χv )

u∈supp+ (x−y) v∈supp− (x−y)

u∈supp+ (x−y) v∈supp− (x−y)

(∀ x, y ∈ dom f with f (x) > f (y)). (6.97) Proof. The proof is much the same as that for Theorem 6.67. Global optimality (minimality) for a quasi M-convex function is characterized by local optimality. Theorem 6.76 (Quasi M-optimality criterion). (1) For f : ZV → R ∪ {+∞} satisfying (QMw ) and x ∈ dom f , we have f (x) < f (y)

(∀ y ∈ ZV \ {x}) ⇐⇒ Δf (x; v, u) > 0

(∀ u, v ∈ V, u = v).

(2) For f : ZV → R ∪ {+∞} satisfying (SSQM = w ) and x ∈ dom f , we have f (x) ≤ f (y)

(∀ y ∈ ZV ) ⇐⇒ Δf (x; v, u) ≥ 0

(∀ u, v ∈ V ).

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Proof. It suffices to show ⇐. (1) is immediate from (6.94) in Theorem 6.67 and (2) is from (6.97) in Theorem 6.75. The minimizer cut theorem (Theorem 6.28) for M-convex functions can be generalized for quasi M-convex functions. Theorem 6.77 (Quasi M-minimizer cut). Let f : ZV → R ∪ {+∞} be a function with (SSQM = ), and assume arg min f = ∅. Then (1), (2), and (3) in Theorem 6.28 hold true. Proof. The proof of Theorem 6.28 is valid here when (M-EXC[Z]) is replaced with (SSQM = ). The proximity theorem for M-convex functions (Theorem 6.37) can be generalized for quasi M-convex functions. Theorem 6.78 (Quasi M-proximity theorem). Let f : ZV → R ∪ {+∞} be a function with (SSQM = ), n = |V |, and α ∈ Z++ . If xα ∈ dom f satisfies (6.66), then arg min f = ∅ and there exists x∗ ∈ arg min f with (6.67). Proof. It suffices to show that, for any γ ∈ R with γ > inf f , there exists some x∗ ∈ dom f satisfying f (x∗ ) ≤ γ and (6.67). Suppose that x∗ ∈ dom f minimizes ||x∗ −xα ||1 among all vectors satisfying f (x∗ ) ≤ γ. In the following, we fix v ∈ V and prove xα (v) − x∗ (v) ≤ (n − 1)(α − 1). (The inequality x∗ (v) − xα (v) ≤ (n − 1)(α − 1) can be shown similarly.) We may assume xα (v) > x∗ (v). Put ) α ⎫ ⎧ ) x (v) ≥ y(v) ≥ x∗ (v) ⎪ ⎪ ) ⎪ ⎪ ⎨ ) xα (w) ≤ y(w) ≤ x∗ (w) (∀ w ∈ supp− (xα − x∗ )) ⎬ . S = y ∈ dom f )) α x (u) = y(u) (∀ u ∈ V \ ({v} ∪ supp− (xα − x∗ )) ⎪ ⎪ ) ⎪ ⎪ ⎭ ⎩ ) xα (V ) = y(V ) Claim 1: For y ∈ arg min{f (y  ) | y  ∈ S}, we have y(v) = x∗ (v). (Proof of Claim 1) Suppose that y(v) > x∗ (v). From the definition of x∗ we have f (y) > f (x∗ ). By (SSQM = ) for y, x∗ , and v ∈ supp+ (y−x∗ ) ⊆ supp+ (xα −x∗ ), there exists w ∈ supp− (y − x∗ ) ⊆ supp− (xα − x∗ ) such that if Δf (x∗ ; v, w) > 0 then Δf (y; w, v) < 0. By the choice of x∗ , we have Δf (x∗ ; v, w) > 0 and hence f (y − χv + χw ) < f (y). Since y − χv + χw ∈ S, this contradicts the minimality of f (y). Thus Claim 1 is proved. Take any y from arg min{f (y  ) | y  ∈ S}, and represent it as  μw χ w . y = xα − λχv + w∈supp− (xα −x∗ )

We have λ = xα (v) − x∗ (v) by Claim 1. Claim 2: For any w ∈ supp− (xα − x∗ ) with μw > 0 and μ ∈ [0, μw − 1]Z , f (xα − (μ + 1)(χv − χw )) < f (xα − μ(χv − χw )).

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(Proof of Claim 2) We prove the claim by induction on μ. For μ ∈ [0, μw − 1]Z , put x = xα − μ(χv − χw ), and assume x ∈ dom f . Note that x ∈ S and x (v) > x∗ (v), and hence Claim 1 implies f (x ) > f (y). Since supp− (y−x ) = {v}, (SSQM = ) for y, x , and w ∈ supp+ (y−x ) implies that if Δf (y; v, w) > 0 then Δf (x ; w, v) < 0. By Claim 1 we have Δf (y; v, w) > 0, from which Claim 2 follows. Claim 2 and (6.66) imply f (xα − μw (χv − χw )) < · · · < f (xα − (χv − χw )) < f (xα ) ≤ f (xα − α(χv − χw )) for any w with μw > 0. Hence, μw ≤ α − 1 for any w ∈ supp− (xα − x∗ ), and  μw ≤ (n − 1)(α − 1), xα (v) − x∗ (v) = xα (v) − y(v) = λ = w∈supp− (xα −x∗ )

where the third equality follows from xα (V ) = y(V ). Theorem 6.79 (Quasi M-minimizer cut with scaling). Let f : ZV → R ∪ {+∞} be a function satisfying (SSQM = ) with arg min f = ∅, and assume α ∈ Z++ and n = |V |. Then (1) and (2) in Theorem 6.39 hold true. Proof. We prove (2), while (1) can be proved similarly. Put xα = x + α(χv − χu ). We may assume max{x∗ (v) | x∗ ∈ arg min f } < xα (v); otherwise we are done. Let x∗ be an element of arg min f with x∗ (v) maximum. The rest of the proof is the same as the proof of Theorem 6.78 (from Claim 1 until the end).

Bibliographical Notes The concept of M-convex functions was introduced by Murota [137] and that of M -convex functions by Murota–Shioura [151]; Theorems 6.2 and 6.3 are due to [151]. The local exchange axiom (Theorem 6.4) is given in Murota [137], and the weak exchange axiom (Theorem 6.5) is explicit in Murota [147]. Quadratic functions of the form (6.23) are treated in Camerini–Conforti– Naddef [22], and their M -convexity is observed in Murota–Shioura [151]. Proposition 6.8 (characterization of quadratic M-convex functions) is due to Murota– Shioura [155]. Quadratic functions defined by symmetric matrices of the form (6.29), or (6.30), are treated in Hochbaum–Shamir–Shanthikumar [92], and their M-convexity is noted by A. Shioura. Quasi-separable convex functions (6.32) are considered by [22], and their M -convexity is pointed out in [151]. The M -convexity of laminar convex functions (6.34) and minimum-value functions (6.36) is due to Danilov–Koshevoy–Murota [34], [35]. The names laminar convex functions and minimum-value functions are coined in this book. The basic operations in section 6.4 are listed in Murota [141], [144], [147]. Theorem 6.13 (8) (infimal convolution) is due to Murota [137], whereas Theorem 6.15 (2) (projection of M -convex functions) is due to [141]. The scaling operation for M-convex functions is considered by Moriguchi–Murota–Shioura [133].

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The supermodularity of M -convex functions (Theorem 6.19) is observed by Murota–Shioura [153]. A special case for valuated matroids was noted earlier by Dress–Terhalle [40] and Murota [138]. Theorem 6.24 (descent direction) is observed by Murota–Tamura [160] as a generalization of its special (but essential) case with dom f ⊆ {0, 1}V due to Fujishige–Yang [69]. Proposition 6.25 is in Murota [137], [140], [142]. Theorems on minimizers are of fundamental importance. Theorem 6.26 (Moptimality criterion) and Theorem 6.30 (characterization by minimizers) are by Murota [137]. Theorem 6.28 (M-minimizer cut) is by Shioura [190]. The connection to the gross substitutes property was studied almost simultaneously by Danilov–Koshevoy–Lang [33], Fujishige–Yang [69], and Murota–Tamura [160]. The concave version of (6.60) is identical to condition GS in [33]. Propositions 6.32 and 6.33 and Theorem 6.34 are due to [160], and Proposition 6.35 and Theorem 6.36 are due to [33]. Results about minimizers under scaling are relatively new. Theorem 6.37 (M-proximity theorem) is by Moriguchi–Murota–Shioura [133]. Theorem 6.39 (Mminimizer cut with scaling) is by Tamura [197]. The convex extension of M-convex functions has been understood step by step. Convex extensibility (latter half of Theorem 6.42) and the characterization by minimizers (Theorem 6.43) are in Murota [137]. Integral convexity (Theorem 6.42) and convex extension for a pair of M-convex functions (Theorem 6.44) are by Murota–Shioura [153]. Polyhedral M-convex functions are investigated by Murota–Shioura [152], to which all the theorems in section 6.11 (Theorems 6.45, 6.47, 6.48, 6.49, 6.50, 6.51, 6.52) as well as Proposition 6.53 are ascribed. M-convexity for nonpolyhedral convex functions is considered in Murota–Shioura [156], [157]. The correspondence between positively homogeneous M-convex functions and distance functions (Theorem 6.59) is established for the case of Z in Murota [141] and generalized to the case of R in Murota–Shioura [152]. Proposition 6.56 is stated in Murota [147]. Theorem 6.61 for directional derivatives and subgradients is shown for the case of Z in Murota [140], [141] and generalized to the case of R in Murota–Shioura [152]. Theorem 6.63 (characterizations in terms of directional derivatives, subdifferentials, and minimizers) is by [152], whereas its ramification with integrality (Theorem 6.64) is stated in Murota [147]. The concept of quasi M-convex functions was introduced by Murota–Shioura [154], to which almost all major results in section 6.14 (Theorems 6.67, 6.68, 6.72, 6.75, 6.76, 6.77, 6.78) are ascribed. Exceptions are Theorem 6.74 (characterization of M-convex functions by level sets) by Shioura [191] and Theorem 6.79 (quasi M-minimizer cut with scaling) by Tamura [197]. Zimmermann [221] considers combinatorial optimization problems with quasi-convex objective functions in real variables. M-convex functions find applications in resource allocation problems (Katoh– Ibaraki [110], Moriguchi–Shioura [134]), mathematical economics (to be treated in Chapter 11), and analysis of polynomial matrices (to be treated in Chapter 12).

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Chapter 7

L-Convex Functions

L-convex functions form another class of well-behaved discrete convex functions. They are defined in terms of an abstract axiom involving submodularity and are characterized as functions obtained by piecing together L-convex sets in a consistent way or as collections of submodular set functions with some consistency. Fundamental properties of L-convex functions are established in this chapter, including the local optimality criterion for global optimality, the proximity theorem for minimizers, discrete midpoint convexity, integral convexity, and extensibility to convex functions. Duality and conjugacy issues are treated in Chapter 8 and algorithms in Chapter 10.

7.1

L-Convex Functions and L -Convex Functions

We recall the definitions of L-convex functions and L -convex functions from section 1.4.1. A function g : ZV → R ∪ {+∞} with dom g = ∅ is said to be an L-convex function if it satisfies (SBF[Z]) (TRF[Z])

g(p) + g(q) ≥ g(p ∨ q) + g(p ∧ q) (∀ p, q ∈ ZV ), ∃ r ∈ R such that g(p + 1) = g(p) + r (∀ p ∈ ZV ).

(SBF[Z]) is submodularity on the integer lattice and (TRF[Z]) linearity in the direction of 1. Note that we have r ∈ Z if g is integer valued. Also recall the submodularity inequality g(p) + g(q) ≥ g(p ∨ q) + g(p ∧ q).

(7.1)

We denote by L[Z → R] the set of L-convex functions and by L[Z → Z] the set of integer-valued L-convex functions. Since an L-convex function g is linear in the direction of 1, we may dispense with this direction as far as we are concerned with its nonlinear behavior. Namely, instead of the function g in n = |V | variables, we may consider the restriction g  to 177

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Chapter 7. L-Convex Functions

an arbitrarily chosen coordinate plane, say, p(u0 ) = 0 for some u0 ∈ V , where the restriction g  is a function in n − 1 variables defined by g  (p ) = g(0, p ), 

with the notation V  = V \ {u0 } and (p0 , p ) ∈ Z × ZV . A function derived from an L-convex function by such a restriction is called an L -convex function. More formally, an L -convex function is defined as follows. Let 0 denote a new element not in V , and put V˜ = {0} ∪ V . A function g : ZV → R ∪ {+∞} is called ˜ L -convex if the function g˜ : ZV → R ∪ {+∞} defined by g˜(p0 , p) = g(p − p0 1)

(p0 ∈ Z, p ∈ ZV )

(7.2)

is an L-convex function. Note that g˜ satisfies (TRF[Z]) with r = 0. We denote by L [Z → R] the set of L -convex functions and by L [Z → Z] the set of integer-valued L -convex functions. It turns out that L -convexity can be characterized by a kind of generalized submodularity: (SBF [Z]) g(p) + g(q) ≥ g((p − α1) ∨ q) + g(p ∧ (q + α1)) (∀ p, q ∈ ZV , ∀ α ∈ Z+ ), which we call translation submodularity. Note that α is restricted to be nonnegative and this inequality for α = 0 agrees with the original submodularity (SBF[Z]). Theorem 7.1. For a function g : ZV → R ∪ {+∞} with dom g = ∅, we have g is L -convex ⇐⇒ g satisfies (SBF [Z]). Proof. Let g˜ be defined by (7.2). The submodularity of g˜, i.e., g˜(p0 , p) + g˜(q0 , q) ≥ g˜(p0 ∨ q0 , p ∨ q) + g˜(p0 ∧ q0 , p ∧ q), is translated to a condition g(p − p0 1) + g(q − q0 1) ≥ g((p ∨ q) − (p0 ∨ q0 )1) + g((p ∧ q) − (p0 ∧ q0 )1) on g. Assuming α = q0 − p0 ≥ 0, put p = p − p0 1 and q  = q − q0 1. Then (p ∨ q) − (p0 ∨ q0 )1 = (p − α1) ∨ q  and (p ∧ q) − (p0 ∧ q0 )1 = p ∧ (q  + α1). Hence the above inequality is equivalent to (SBF [Z]). L -convex functions are conceptually equivalent to L-convex functions, but the class of L -convex functions is larger than that of L-convex functions. The condition (7.3) below is stronger than (SBF [Z]) in that it requires the inequality not only for nonnegative α but also for negative α. Theorem 7.2. An L-convex function g ∈ L[Z → R] satisfies g(p) + g(q) ≥ g((p − α1) ∨ q) + g(p ∧ (q + α1))

(p, q ∈ ZV , α ∈ Z).

(7.3)

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7.1. L-Convex Functions and L -Convex Functions

179

Proof. By (SBF[Z]) and (TRF[Z]) we see g(p) + g(q) = g(p − α1) + g(q) + αr ≥ g((p − α1) ∨ q) + g((p − α1) ∧ q) + αr = g((p − α1) ∨ q) + g(p ∧ (q + α1)). Theorem 7.3. An L-convex function is L -convex. Conversely, an L -convex function is L-convex if and only if it satisfies (TRF[Z]). Proof. This follows from Theorem 7.1 and the obvious implications (7.3) ⇒ (SBF [Z]) ⇒(SBF[Z]). For ease of reference we summarize the relationship between L and L as Ln ⊂ Ln  Ln+1 ,

(7.4)

where Ln and Ln denote, respectively, the sets of L-convex functions and L -convex functions in n variables, and the expression Ln  Ln+1 means a correspondence of their elements (functions) up to the constant r in (TRF[Z]), where (7.2) gives the correspondence under the normalization of r = 0. By the equivalence between L-convex functions and L -convex functions all theorems stated for L-convex functions can be rephrased for L -convex functions, and vice versa. In this book we primarily work with L-convex functions, making explicit statements for L -convex functions when appropriate. A set function ρ : 2V → R ∪ {+∞} can be identified with a function g : ZV → R ∪ {+∞} with dom g ⊆ {0, 1}V through  ρ(X) (p = χX , X ⊆ V ), g(p) = (7.5) +∞ (otherwise). The following states that the submodularity of ρ is the same as the L -convexity of g. Proposition 7.4. Let ρ : 2V → R ∪ {+∞} be a set function with dom ρ = ∅ and g : ZV → R ∪ {+∞} be the associated function in (7.5). Then we have ρ is submodular ⇐⇒ g is L -convex. Proof. (SBF [Z]) for α = 0 is equivalent to the submodularity (4.9) of ρ. (SBF [Z]) for α ≥ 1 is void for a function g with dom g ⊆ {0, 1}V since (p − α1) ∨ q = q and p ∧ (q + α1) = p for any p, q ∈ {0, 1}V and α ≥ 1. The proposition above shows that L -convex functions effectively contain submodular set functions as a subclass; i.e., S[R] → L [Z → R], where → denotes the embedding by (7.5).

S[Z] → L [Z → Z],

(7.6)

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Chapter 7. L-Convex Functions

We mention here the following fundamental fact, showing that submodularity (on the integer lattice) is in fact a local property. The proof is easy and omitted. Proposition 7.5 (Local submodularity). Let g : ZV → R ∪ {+∞} be a function with dom g being L -convex. Then g satisfies the submodularity inequality (7.1) for all p, q ∈ ZV if and only if it satisfies (7.1) for all p, q ∈ ZV with ||p − q||∞ = 1. Note 7.6. With an M -concave function h : ZV → R ∪ {−∞} with dom h = {0, 1}V , we may associate a function g : ZV → R∪{+∞} such that dom g = {0, 1}V and g(p) = h(p) for p ∈ {0, 1}V . Since h is submodular by Theorem 6.19, g is L -convex by Proposition 7.4. In this sense we have M -concave ⇒ L -convex for functions on {0, 1}-vectors. The converse does not hold, as is demonstrated by an L -convex function g : Z3 → R ∪ {+∞} with dom g = {0, 1}3 defined by g(1, 1, 1) = −2, g(1, 1, 0) = g(1, 0, 1) = −1, and g(0, 0, 0) = g(1, 0, 0) = g(0, 1, 0) = g(0, 0, 1) = g(0, 1, 1) = 0. Note that h = −∞ and g = +∞ outside {0, 1}V and that any function on {0, 1}V can be extended to a convex function and to a concave function.

7.2

Discrete Midpoint Convexity

We show a characterization of L -convexity in terms of discrete midpoint convexity     p+q p+q g(p) + g(q) ≥ g (7.7) +g (p, q ∈ ZV ), 2 2 which is an obvious approximation to the midpoint convexity (1.3) of ordinary convex functions; see Fig. 7.1. We consider another property for a function g: (L -APR[Z]) For any p, q ∈ ZV with supp+ (p − q) = ∅, g(p) + g(q) ≥ g(p − χX ) + g(q + χX ), where X = arg max{p(v) − q(v)}. v∈V

This says that the sum of the function values at a pair of points (p, q) does not increase when the pair is replaced with another pair (p − χX , q + χX ) of closer points. Theorem 7.7. For a function g : ZV → R ∪ {+∞} with dom g = ∅, we have (SBF [Z]) ⇐⇒ (L -APR[Z]) ⇐⇒ discrete midpoint convexity (7.7). Hence, each of these is a necessary and sufficient condition for g to be L -convex. Proof. First, Theorem 7.1 shows the equivalence of (SBF [Z]) to L -convexity. [(SBF [Z])⇒(L -APR[Z])]: Suppose that supp+ (p − q) = ∅ and put α = maxv∈V {p(v) − q(v)} − 1. We have α ≥ 0, (p − α1) ∨ q = q + χX , and p ∧ (q + α1) = p − χX . Hence (L -APR[Z]) follows from (SBF [Z]).

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7.3. Examples

181

q q

p+q

p+q

2

2

p+q p

p+q 2

p+q

2

2

q

p+q p

p

2

Figure 7.1. Discrete midpoint convexity.

and q  = p+q and define p , q  by [(L -APR[Z])⇒(7.7)]: Put p = p+q 2 2     p (v) (p(v) ≥ q(v)), q (v) (p(v) ≥ q(v)),   p (v) = q (v) = q  (v) (p(v) ≤ q(v)), p (v) (p(v) ≤ q(v)). We have |p (v) − q  (v)| ≤ 1 (v ∈ V ), supp+ (p − q  ) ⊆ supp+ (p − q), and supp− (p − q  ) ⊆ supp− (p − q). Starting with (p, q) and applying (L -APR[Z]) repeatedly, we obtain g(p) + g(q) ≥ g(p ) + g(q  ). Applying (L -APR[Z]) to (p , q  ) yields g(p ) + g(q  ) ≥ g(p ) + g(q  ). Hence follows g(p) + g(q) ≥ g(p ) + g(q  ). [(7.7)⇒(SBF [Z])]: (SBF [Z]) for g is equivalent to the submodularity of g˜ in (7.2) (cf. proof of Theorem 7.1). Since dom g is an L -convex set by (7.7) and (5.15), dom g˜ is an L-convex set. By Proposition 7.5, the submodularity of g˜ is equivalent to the local submodularity of g˜ and the latter holds if and only if g(p + χX ) + g(p + χY ) ≥ g(p) + g(p + χX + χY ), g(p + χX − 1) + g(p + χY ) ≥ g(p) + g(p + χX + χY − 1) for all p ∈ ZV and X, Y ⊆ V with X ∩ Y = ∅. These two conditions follow easily from discrete midpoint convexity (7.7).

7.3

Examples

We have already seen L-convexity in network flows and in matroids (section 2.2, section 2.4). In this section we see some other examples of L-convex functions, such as linear functions, quadratic functions, and separable convex functions. First we note the following facts. Proposition 7.8. (1) The effective domain of an L-convex function is an L-convex set. (2) The effective domain of an L -convex function is an L -convex set. Proof. (1) (SBF[Z]) and (TRF[Z]) for g imply (SBS[Z]) and (TRS[Z]) for D = dom g. (2) Similarly, (SBF [Z]) for g implies (SBS [Z]) for D = dom g.

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182 Linear functions

Chapter 7. L-Convex Functions A linear (or affine) function48 g(p) = α + p, x

(p ∈ dom g)

(7.8)

with x ∈ Rn and α ∈ R is L-convex or L -convex according as dom g is L-convex or L -convex. Quadratic functions

A quadratic function g(p) =

n  n 

aij p(i)p(j)

(p ∈ Zn )

(7.9)

i=1 j=1

with aij = aji ∈ R (i, j = 1, . . . , n) is L -convex if and only if aij ≤ 0 (i = j),

n 

aij ≥ 0

(i = 1, . . . , n),

(7.10)

j=1

which can be proved as in Theorem 2.7. Accordingly, g is L-convex if and only if aij ≤ 0 (i = j),

n 

aij = 0

(i = 1, . . . , n).

(7.11)

j=1

In Example 2.1 (Poisson equation) and Example 2.2 (electrical network), we seen the matrices ⎡ ⎤ ⎡ 2 −1 −g1 −g2 −g5 g1 + g2 + g5 ⎢ −1 ⎥ ⎢ 2 −1 −g g + g 0 −g4 1 1 4 ⎢ ⎥, ⎢ ⎣ ⎣ −1 2 −1 ⎦ −g2 0 g2 + g3 −g3 −1 2 −g5 −g4 −g3 g3 + g4 + g5

have ⎤ ⎥ ⎥, ⎦

which satisfy (7.10) and (7.11), respectively. Separable convex functions g(p) =

A separable convex function n 

gi (p(i))

(p ∈ dom g)

(7.12)

i=1

with univariate discrete convex functions gi ∈ C[Z → R] (i = 1, . . . , n) is L -convex if dom g is an L -convex set.49 In particular, a separable convex function with a chain condition  n i=1 gi (p(i)) (p(1) ≤ p(2) ≤ · · · ≤ p(n)) (7.13) g(p) = (p ∈ Zn ) +∞ (otherwise) 48 In this section, V = {1, . . . , n}, g denotes a real-valued function in integer variables, i.e., g : Zn → R ∪ {+∞}, and p(i) is the ith component of an integer vector p = (p(i) | i = 1, . . . , n) ∈ Zn . 49 It is easy to verify discrete midpoint convexity (7.7) for g.

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183

is L -convex. For gij ∈ C[Z → R] (i = j; i, j = 1, . . . , n), the function  gij (p(i) − p(j)) (p ∈ dom g) g(p) =

(7.14)

i =j

is L-convex if dom g is an L-convex set. As special cases of (7.12) and (7.14) we see the following. Proposition 7.9. Let ψ ∈ C[Z → R] be a univariate discrete convex function. (1) ψ is L -convex. (2) The function g : Z2 → R ∪ {+∞} defined by g(p) = ψ(p(1) − p(2)) is L-convex. Maximum-component functions

The function

g(p) = max{p(1), . . . , p(n)}

(p ∈ Zn ),

(7.15)

which gives the maximum value of the components of p, is L-convex. Multimodular functions A function h : Zn → R ∪ {+∞} is said to be multimodular if the function g˜ : Zn+1 → R ∪ {+∞} defined by g˜(p0 , p) = h(p(1) − p0 , p(2) − p(1), . . . , p(n) − p(n − 1))

(p0 ∈ Z, p ∈ Zn ) (7.16)

is submodular. This means that a function h : Zn → R ∪ {+∞} is multimodular if and only if it can be represented as h(p) = g(p(1), p(1) + p(2), . . . , p(1) + · · · + p(n))

(p ∈ Zn )

(7.17)

for some L -convex function g. Submodular set functions Any submodular set function may be regarded as an L -convex function by Proposition 7.4.

7.4

Basic Operations

Basic operations on L-convex functions are presented here, whereas the most important operation, transformation by networks, is treated later in section 9.6. L-convex functions admit the following operations. See (6.41) for the definition of the projection g U . Theorem 7.10. Let g, g1 , g2 ∈ L[Z → R] be L-convex functions. (1) For λ ∈ R++ , λg is L-convex. (2) For a ∈ ZV and β ∈ Z \ {0}, g(a + βp) is L-convex in p. (3) For x ∈ RV , g[−x] is L-convex. (4) For U ⊆ V , the projection g U is L-convex provided g U > −∞. (5) For ψv ∈ C[Z → R] (v ∈ V ), , +  g˜(p) = inf g(q) + ψv (p(v) − q(v)) (p ∈ ZV ) q∈ZV

v∈V

(7.18)

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Chapter 7. L-Convex Functions

is L-convex provided g˜ > −∞. (6) The sum g1 + g2 is L-convex provided dom (g1 + g2 ) = ∅. Proof. (1), (2), (3), and (6) are obvious. (5) (TRF[Z]) is easy to see. For (SBF[Z]) we indicate the idea by assuming that, for each p1 , p2 ∈ dom g˜, the infimum in (7.18) is attained by some q1 , q2 ∈ ZV . Proposition 7.9 (2) and (SBF[Z]) for g respectively show ψv (p1 (v) − q1 (v)) + ψv (p2 (v) − q2 (v)) ≥ ψv ([p1 (v) ∨ p2 (v)] − [q1 (v) ∨ q2 (v)]) + ψv ([p1 (v) ∧ p2 (v)] − [q1 (v) ∧ q2 (v)]), g(q1 ) + g(q2 ) ≥ g(q1 ∨ q2 ) + g(q1 ∧ q2 ). On the other hand, (7.18) implies  ψv ([p1 (v) ∨ p2 (v)] − [q1 (v) ∨ q2 (v)]) ≥ g˜(p1 ∨ p2 ), g(q1 ∨ q2 ) + v∈V

g(q1 ∧ q2 ) +



ψv ([p1 (v) ∧ p2 (v)] − [q1 (v) ∧ q2 (v)]) ≥ g˜(p1 ∧ p2 ).

v∈V

Adding these inequalities yields (SBF[Z]) for g˜. (4) A special case of (5) with ψv = δ{0} (v ∈ U ) and ψv = δZ (v ∈ V \ U ) ˆ = shows the L-convexity of g˜ = g2Z δUˆ , where δUˆ is the indicator function of U V U {p ∈ Z | p(v) = 0 (v ∈ U )}. By (6.45), g is the restriction of g˜ to U . Then (SBF[Z]) of g U is immediate from that of g˜, and (TRF[Z]) of g U follows from that of g˜ since g˜(p + χV ) = g˜(p + χU ). Note in Theorem 7.10 (2) that we have a scaling factor β, in contrast to the similar statement (Theorem 6.13 (2)) for M-convex functions. Also note that g˜ in Theorem 7.10 (5) is the infimal convolution of g with a separable convex function. Operations in Theorem 7.10 are also valid for L -convex functions. In addition, restrictions are allowed for L -convex functions. See (3.55) and (6.40) for the definitions of the restrictions g[a,b] and gU . Theorem 7.11. Let g, g1 , g2 ∈ L [Z → R] be L -convex functions. (1) Operations (1)–(6) of Theorem 7.10 are valid for L -convex functions. (2) For a, b ∈ (Z ∪ {±∞})V , the restriction g[a,b] to the integer interval [a, b]  is L -convex provided dom g[a,b] = ∅. (3) For U ⊆ V , the restriction gU is L -convex provided dom gU = ∅. Proof. (2), (3) It is easy to verify (SBF [Z]) for g[a,b] and gU .

Note 7.12. The infimal convolution of two L-convex functions is not necessarily L-convex, and similarly for the infimal convolution of two L -convex functions. Such functions are studied in section 8.3 under the names L2 -convex functions and L2 convex functions, respectively.

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185

Note 7.13. The proviso g U > −∞ in Theorem 7.10 (4) can be weakened to g U (p0 ) > −∞ for some p0 , and similarly for g˜ > −∞ in Theorem 7.10 (5).

7.5

Minimizers

Global optimality for an L-convex function is characterized by local optimality. Theorem 7.14 (L-optimality criterion). (1) For an L-convex function g ∈ L[Z → R] and p ∈ dom g, we have  g(p) ≤ g(p + χY ) (∀ Y ⊆ V ), V g(p) ≤ g(q) (∀ q ∈ Z ) ⇐⇒ g(p) = g(p + 1).

(7.19)

(2) For an L -convex function g ∈ L [Z → R] and p ∈ dom g, we have g(p) ≤ g(q)

(∀ q ∈ ZV ) ⇐⇒ g(p) ≤ g(p ± χY )

(∀ Y ⊆ V ).

(7.20)

Proof. It suffices to prove ⇐ in (1) and (2). We first consider (2). For any disjoint Y, Z ⊆ V , condition (7.20) together with submodularity yields g(p) + g(p + χY − χZ ) ≥ g(p + χY ) + g(p − χZ ) ≥ 2g(p), which implies the optimality criterion (3.65) for integrally convex functions. Since an L -convex function is integrally convex (to be shown in Theorem 7.20), Theorem 3.21 establishes ⇐ in (2). Next, (1) follows from (2), since an L-convex function is L -convex (Theorem 7.3) and the right-hand side of (7.19) implies g(p − χY ) = g(p + χV \Y ) ≥ g(p). The well-known optimality criterion for a submodular set function is an immediate corollary of (2) above. Theorem 7.15. Let ρ be a submodular set function. A subset X ∈ dom ρ is a minimizer of ρ if and only if ρ(X) ≤ ρ(Y ) for any Y that includes X or is included in X. Proof. Let g be the L -convex function associated with ρ (see (7.5) and Proposition 7.4), and apply Theorem 7.14 (2) with p = χX . Although Theorem 7.14 affords a local criterion for global optimality of a point p, a straightforward verification of (7.19) requires O(2n ) function evaluations. The verification can be done in polynomial time as follows. We consider a submodular set function ρp defined by ρp (Y ) = g(p + χY ) − g(p) and note that (7.19) is equivalent to saying that ρp achieves its minimum at Y = ∅. This condition can be verified in polynomial time by the submodular function minimization algorithms in section 10.2. The minimizers of an L-convex function form an L-convex set, a property that is essential for a function to be L-convex.

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186

Chapter 7. L-Convex Functions

Proposition 7.16. For an L-convex function g ∈ L[Z → R], arg min g is an L-convex set if it is not empty. Proof. Suppose D = arg min g is nonempty. We have r = 0 in (TRF[Z]) and hence D satisfies (TRS[Z]). For p, q ∈ D, we have p ∨ q, p ∧ q ∈ D by (SBF[Z]). This shows (SBS[Z]) for D. The following theorem reveals that L-convex functions are characterized as functions obtained by piecing together L-convex sets in a consistent way. This shows how the concept of L-convex functions can be defined from that of L-convex sets. Theorem 7.17. Let g : ZV → R ∪ {+∞} be a function with a bounded nonempty effective domain. (1) g is L-convex ⇐⇒ arg min g[−x] is an L-convex set for each x ∈ RV . (2) g is L -convex ⇐⇒ arg min g[−x] is an L -convex set for each x ∈ RV . Proof. It suffices to prove (1). The implication ⇒ is immediate from Theorem 7.10 (3) and Proposition 7.16. The converse is shown later in Note 7.47.

7.6

Proximity Theorem

We show a proximity theorem for L-convex function minimization, stating that a global optimum of an L-convex function g exists in a neighborhood of a local optimum of its scaling g α defined by g α (p) = g(αp)/α. Note that g α is L-convex by Theorem 7.10 (2), and accordingly, a local minimizer of g α is a global minimizer of g α by the L-optimality criterion (Theorem 7.14). Theorem 7.18 (L-proximity theorem). Assume α ∈ Z++ and n = |V |. (1) Let g : ZV → R ∪ {+∞} be an L-convex function with g(p) = g(p + 1) (∀ p ∈ ZV ). If pα ∈ dom g satisfies g(pα ) ≤ g(pα + αχY )

(∀ Y ⊆ V ),

(7.21)

then arg min g = ∅ and there exists p∗ ∈ arg min g with pα ≤ p∗ ≤ pα + (n − 1)(α − 1)1.

(7.22)

(2) Let g : ZV → R ∪ {+∞} be an L -convex function. If pα ∈ dom g satisfies g(pα ) ≤ g(pα ± αχY )

(∀ Y ⊆ V ),

(7.23)

then arg min g = ∅ and there exists p∗ ∈ arg min g with pα − n(α − 1)1 ≤ p∗ ≤ pα + n(α − 1)1.

(7.24)

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Proof. (1) It suffices to show that, for any β > inf g, there exists p∗ that satisfies g(p∗ ) ≤ β and (7.22); note that there exist only a finite number of p∗ satisfying (7.22). We may assume pα = 0. By (TRF[Z]) with r = 0 there exists p∗ such that g(p∗ ) ≤ β and p∗ ≥ 0; let p∗ be minimal (with respect to order ≥) among such vectors. We have p∗ (v) = 0 for some v ∈ V and (∀ X ⊆ supp+ (p∗ )). (7.25) g(p∗ − χX ) > g(p∗ ) k We can represent p∗ as p∗ = i=1 μi χXi , where μi ∈ Z++ (i = 1, . . . , k), ∅ = ⊂ ⊂ ⊂ X1 = X2 = · · · = Xk = V , and 0 ≤ k ≤ n − 1. Claim 1: 9 8j−1 9 8j−1   μi χXi + μχXj > g μi χXi + (μ + 1)χXj (1 ≤ j ≤ k, 0 ≤ μ ≤ μj − 1). g i=1

i=1

j−1 (Proof of Claim 1) Put p = i=1 μi χXi +μχXj and suppose p ∈ dom g. By Xj ⊆ supp+ (p∗ ) and (7.25) we have g(p∗ −χXj ) > g(p∗ ). Since Xj = arg maxv∈V {p∗ (v) − p(v)}, (L -APR[Z]) shows that g(p∗ − χXj ) > g(p∗ ) ⇒ g(p + χXj ) < g(p). Note that g satisfies (L -APR[Z]) by Theorem 7.7. Claim 2: g(μχXj ) > g((μ + 1)χXj ) (1 ≤ j ≤ k, 0 ≤ μ ≤ μj − 1). j (Proof of Claim 2) Put p = i=1 μi χXi and q = μχXj and suppose q ∈ dom g. Since V \ Xj = arg maxv∈V {q(v) − p(v)} and g(p + χV \Xj ) = g(p − χXj ) > g(p) by Claim 1, (L -APR[Z]) implies g(q) > g(q − χV \Xj ) = g(q + χXj ). It follows from Claim 2 and (7.21) that μi < α for i = 1, . . . , k, and hence 0 ≤ p∗ ≤ (α − 1)

k 

χXi ≤ (n − 1)(α − 1)1.

i=1

(2) This follows from (1) applied to g˜ in (7.2). The algorithmic use of the above theorem is shown in sections 10.3.2 and 10.4.5.

7.7

Convex Extension

This section establishes one of the major properties of L-convex functions, which is that they can be extended to convex functions in real variables. The extensibility to convex functions is by no means obvious from the definition of L-convex functions in terms of axioms referring only to function values on integer points. The convex extension of an L-convex function can be obtained by piecing together the Lov´ asz extensions. With a function g : ZV → R ∪ {+∞} and a point p ∈ dom g we associate a set function ρg,p : 2V → R ∪ {+∞} defined by ρg,p (X) = g(p + χX ) − g(p)

(X ⊆ V ).

(7.26)

If g is L-convex, the associated set function ρg,p belongs to S[R] (i.e., ρg,p is submodular, ρg,p (∅) = 0, and ρg,p (V ) < +∞).

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Chapter 7. L-Convex Functions

The next theorem shows that an L-convex function g can be extended to a convex function, and that the convex extension can be constructed from the Lov´ asz extension of ρg,p for varying p ∈ ZV . Theorem 7.19. Let g ∈ L[Z → R] be an L-convex function and g be its convex closure. (1) For p ∈ dom g and q ∈ [0, 1]R , we have g(p + q) = g(p) + ρˆg,p (q) = g(p) +

m−1 

(ˆ qi − qˆi+1 )(g(p + χUi ) − g(p)) + qˆm (g(p + χUm ) − g(p)),

i=1

(7.27)

where ρˆg,p denotes the Lov´ asz extension (4.6) of the associated set function ρg,p in (7.26), qˆ1 > qˆ2 > · · · > qˆm are the distinct values of the components of q, and Ui = Ui (q) = {v ∈ V | q(v) ≥ qˆi }

(i = 1, . . . , m).

(7.28)

(2) For p ∈ ZV and q ∈ [0, 1]R , we have 50 g(p + q) = (1 − qˆ1 )g(p) +

m−1 

(ˆ qi − qˆi+1 )g(p + χUi ) + qˆm g(p + χUm ).

(7.29)

i=1

(3) g(p) = g(p) (p ∈ ZV ). (4) g(q + α1) = g(q) + αr (q ∈ RV , α ∈ R) with the constant r in (TRF[Z]). (5) Expression (7.27) is valid for q ∈ N0 , where N0 = {q ∈ RV | max q(v) − min q(v) ≤ 1}. v∈V

v∈V

Proof. (2) For each p ∈ ZV , let hp (q) denote the function in q ∈ [0, 1]R defined by the right-hand side of (7.29). If p ∈ dom g, ρg,p belongs to S[R], and hence hp is a polyhedral convex function by Theorem 4.16. With the representation of a real vector s = p + q with p = s and q = s − s we define a function h : RV → R ∪ {+∞} by h(s) = hp (q). We have s ∈ dom h ⇐⇒ p + χUi ∈ dom g

(i = 0, 1, . . . , m),

where U0 = ∅ and Ui = Ui (q) for i = 1, . . . , m, as in (7.28). By construction, h is convex in [p, p + 1]R for each p ∈ ZV . Furthermore, it is convex in the entire space RV , because h(s − α1) = h(s) − αr for s ∈ RV and α ∈ R by (TRF[Z]) of g, and for each s ∈ RV there exist α ∈ R and p ∈ ZV such that s − α1 is an interior point of [p, p + 1]R . Obviously, we have h(p) = g(p) for every p ∈ ZV , and h is the maximum among convex functions with this property. Therefore, h = g. (1) If p ∈ dom g, the right-hand side of (7.29) can be rewritten as (7.27). 50 Expression

(7.29) does not involve ∞ − ∞ even for p ∈ / dom g.

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(3) This is a special case of (7.29) with q = 0. (4) This is immediate from (7.29) and (TRF[Z]). (5) For q ∈ N0 we put α = minv∈V q(v). By (4), g(p+q) = g(p+(q−α1))+αr. We can apply (7.27) to g(p + (q − α1)) since q − α1 ∈ [0, 1]R . The above theorem implies the integral convexity of an L-convex function and hence that of an L -convex function. Theorem 7.20. An L -convex function is integrally convex. In particular, an L -convex function is convex extensible. An integrally convex function with submodularity (SBF[Z]) is called a submodular integrally convex function. This turns out to be a synonym of L -convex function. Theorem 7.21. For a function g : ZV → R ∪ {+∞} with dom g = ∅, g is L -convex ⇐⇒ g is submodular integrally convex. Proof. The implication ⇒ follows from (SBF[Z]) and Theorem 7.20. The converse can be shown as follows. By the integral convexity of g, the convex closure g coincides with the local convex extension, and, by the submodularity of g, the latter is obtained as the Lov´ asz extension (7.27) of ρg,p in (7.26). Therefore, we have       p+q p+q p+q 2g =g +g 2 2 2 for any p, q ∈ ZV . On the other hand, we have  g(p) + g(q) = g(p) + g(q) ≥ 2g

p+q 2



from convex extensibility and midpoint convexity (1.3). From these follows the discrete midpoint convexity (7.7) of g, which means L -convexity by Theorem 7.7.

Note 7.22. Submodularity (SBF[Z]) alone does not guarantee convex extensibility. For example, any function g : Z2 → R ∪ {+∞} with dom g = {p ∈ Z2 | p(1) = p(2)} is submodular.

7.8

Polyhedral L-Convex Functions

As we have seen, L-convex functions on the integer lattice can be extended to convex functions in real variables. The convex extension of an L-convex function is a polyhedral convex function when restricted to a finite interval. Motivated by this

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Chapter 7. L-Convex Functions

we define here the concept of L-convexity for polyhedral convex functions in general and show that major properties of L-convex functions survive in this generalization. A polyhedral convex function g : RV → R ∪ {+∞} with domR g = ∅ is said to be L-convex if it satisfies (SBF[R]) g(p) + g(q) ≥ g(p ∨ q) + g(p ∧ q) (∀ p, q ∈ RV ), (TRF[R]) ∃ r ∈ R such that g(p + α1) = g(p) + αr (∀ p ∈ RV , ∀ α ∈ R). (SBF[R]) is submodularity on the real-vector lattice and (TRF[R]) is linearity in the direction of 1. We denote by L[R → R] the set of polyhedral L-convex functions. Polyhedral L-concave functions are defined in an obvious way. Submodularity (SBF[R]) is in fact a local property. Under auxiliary conditions on the effective domain, it is implied by submodularity for a certain set of local pairs of p and q. The following two propositions are mentioned here; the proofs are straightforward and omitted (see Theorems 4.26 and 4.27 of Murota–Shioura [152]). Proposition 7.23. Let g : RV → R ∪ {+∞} be a function with domR g a closed set. Then g is submodular (SBF[R]) if, for each p0 ∈ domR g, there exists ε = ε(p0 ) > 0 such that inequality (7.1) is satisfied for all p, q with ||p − p0 ||∞ ≤ ε and ||q − p0 ||∞ ≤ ε. Proposition 7.24. Let g : RV → R∪{+∞} be a function with domR g an interval. (1) g is submodular (SBF[R]) if g(p + λχu ) + g(p + μχv ) ≥ g(p) + g(p + λχu + μχv )

(7.30)

for all p ∈ domR g; u, v ∈ V with u = v; and λ, μ ∈ R+ . (2) g is submodular (SBF[R]) if (7.30) holds for all p ∈ domR g; u, v ∈ V with u = v; λ ∈ [0, pˆi−1 − pˆi ]R ; and μ ∈ [0, pˆj−1 − pˆj ]R , where pˆ1 > pˆ2 > · · · > pˆm denote the distinct values of the components of p, the indices i and j are such that p(u) = pˆi and p(v) = pˆj , and [0, pˆ0 − pˆ1 ]R means [0, +∞)R by convention. The Lov´asz extension ρˆ of a submodular set function ρ is a polyhedral L-convex function. Proposition 7.25. For ρ ∈ S[R] we have ρˆ ∈ L[R → R]. Proof. (TRF[R]) for g = ρˆ is obvious, and we show (SBF[R]) below. First, assume ρ is finite valued. Then domR ρˆ = RV . By Proposition 7.24 (2), (SBF[R]) follows from inequality (7.30) for u ∈ Ui \ Ui−1 , v ∈ Uj \ Uj−1 , λ ∈ [0, pˆi−1 − pˆi ]R , and μ ∈ [0, pˆj−1 − pˆj ]R , where Ui (i = 1, . . . , m) are defined in (4.4) and U0 = ∅. Expression (4.6) shows g(p + λχu ) = g(p) + λ[ρ(Ui−1 ∪ {u}) − ρ(Ui−1 )], g(p + μχv ) = g(p) + μ[ρ(Uj−1 ∪ {v}) − ρ(Uj−1 )].

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191

If i = j, we have g(p + λχu + μχv ) + g(p) = λ[ρ(Ui−1 ∪ {u}) − ρ(Ui−1 )] + μ[ρ(Uj−1 ∪ {v}) − ρ(Uj−1 )] + 2g(p) = g(p + λχu ) + g(p + μχv ), and, if i = j, we may assume λ ≥ μ and then we have g(p + λχu + μχv ) + g(p) = λ[ρ(Ui−1 ∪ {u}) − ρ(Ui−1 )] + μ[ρ(Ui−1 ∪ {u, v}) − ρ(Ui−1 ∪ {u})] + 2g(p) ≤ λ[ρ(Ui−1 ∪ {u}) − ρ(Ui−1 )] + μ[ρ(Ui−1 ∪ {v}) − ρ(Ui−1 )] + 2g(p) = g(p + λχu ) + g(p + μχv ) from the submodularity of ρ. Hence follows (7.30). The general case where ρ may possibly take the value of +∞ can be reduced to the finite-valued case. For each k ∈ Z++ , we define ρk by ρk (X) = min {ρ(X \ Y ) + k|Y |} Y ⊆X

(X ⊆ V )

and put gk = ρˆk . We have ρk < +∞ and ρk ∈ S[R] and therefore (SBF[R]) for gk . Since ρ(X) = limk→∞ ρk (X) (∀ X ⊆ V ), we have g(p) = limk→∞ gk (p) for each p ∈ RV . Hence follows (SBF[R]) for g. An L-convex function on integer points naturally induces a polyhedral Lconvex function via convex extension (which exists by Theorem 7.20). Theorem 7.26. The convex extension g of an L-convex function g ∈ L[Z → R] on the integer lattice is a polyhedral L-convex function, i.e., g ∈ L[R → R], provided that g is polyhedral. Proof. We use (7.27) in Theorem 7.19. (TRF[R]) is obvious. By Proposition 7.25, ρˆg,p is submodular on RV , and hence g is submodular on [p, p+1]R for each p ∈ ZV . The submodularity (SBF[R]) of g on RV follows from this. Example 7.27. The convex extension g of an L-convex function g ∈ L[Z → R] may consist of an infinite number of linear pieces, in which case g is not polyhedral convex. For example, we have g ∈ L[Z → R] and g ∈ / L[R → R] for g : Z2 → R 2 defined by g(p) = (p(1) − p(2)) . It is worth noting that, if domZ g is bounded, g is polyhedral and therefore g ∈ L[R → R] by Theorem 7.26. Polyhedral L-convex functions with integrality (6.75) are referred to as integral polyhedral L-convex functions, the set of which is denoted by L[Z|R → R]. Polyhedral L-convex functions with dual integrality (6.76) are referred to as dual-integral polyhedral L-convex functions, the set of which is denoted by L[R → R|Z].

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Chapter 7. L-Convex Functions

By Theorem 7.26 and Proposition 7.16 an integral polyhedral L-convex function is nothing but a polyhedral L-convex function that can be obtained as the convex extension of an L-convex function on integer points. Therefore, we have L[Z|R → R] ⊆ L[R → R],

L[Z|R → R] → L[Z → R],

(7.31)

where the second expression means that there exists an injection from L[Z|R → R] to L[Z → R], representing an embedding of L[Z|R → R] into L[Z → R]. The effective domain of a polyhedral L-convex function is an L-convex polyhedron, which is homogeneous in direction 1. Hence, polyhedral L -convex functions can be defined as the restriction of polyhedral L-convex functions, just as L -convex functions on integer points are defined from L-convex functions via (7.2). We denote by L [R → R] the set of polyhedral L -convex functions and by L [Z|R → R] the set of integral polyhedral L -convex functions. The relationship between L and L is described by Ln ⊂ Ln  Ln+1 , where Ln and Ln denote, respectively, the sets of polyhedral L-convex functions and polyhedral L -convex functions in n variables. The R-counterpart of (SBF [Z]) is the following: (SBF [R]) g(p) + g(q) ≥ g((p − α1) ∨ q) + g(p ∧ (q + α1)) (∀ p, q ∈ RV , ∀ α ∈ R+ ). Theorem 7.28. For a polyhedral convex function g : RV → R ∪ {+∞} with domR g = ∅, (SBF [R]) is equivalent to polyhedral L -convexity. The condition (7.32) below is stronger than (SBF [R]) in that it requires the inequality not only for nonnegative α but also for negative α. Theorem 7.29. A polyhedral L-convex function g ∈ L[R → R] satisfies g(p) + g(q) ≥ g((p − α1) ∨ q) + g(p ∧ (q + α1))

(∀ p, q ∈ RV , ∀ α ∈ R).

(7.32)

Theorem 7.30. A polyhedral L-convex function is polyhedral L -convex. Conversely, a polyhedral L -convex function is polyhedral L-convex if and only if it satisfies (TRF[R]). Almost all properties of L-convex functions on integer points carry over to polyhedral L-convex functions. To be specific, Theorems 7.10, 7.11, and 7.14 and Proposition 7.16 are adapted as follows. Note, however, that the proofs are not straightforward adaptations; see Murota–Shioura [152]. Theorem 7.31. Let g, g1 , g2 ∈ L[R → R] be polyhedral L-convex functions. (1) For λ ∈ R++ , λg is polyhedral L-convex. (2) For a ∈ RV and β ∈ R \ {0}, g(a + βp) is polyhedral L-convex in p. (3) For x ∈ RV , g[−x] is polyhedral L-convex. (4) For U ⊆ V , the projection g U is polyhedral L-convex provided g U > −∞.

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7.9. Positively Homogeneous L-Convex Functions (5) For ψv ∈ C[R → R] (v ∈ V ), , +  g˜(p) = inf g(q) + ψv (p(v) − q(v)) q∈RV

193

(p ∈ RV )

(7.33)

v∈V

is polyhedral L-convex provided g˜ > −∞. (6) The sum g1 + g2 is polyhedral L-convex provided dom (g1 + g2 ) = ∅. Theorem 7.32. Let g, g1 , g2 ∈ L [R → R] be polyhedral L -convex functions. (1) Operations (1)–(6) of Theorem 7.31 are valid for polyhedral L -convex functions. (2) For a, b ∈ (R ∪ {±∞})V , the restriction g[a,b] to the real interval [a, b] is polyhedral L -convex provided dom g[a,b] = ∅. (3) For U ⊆ V , the restriction gU is polyhedral L -convex provided dom gU = ∅. Theorem 7.33 (L-optimality criterion). (1) For a polyhedral L-convex function g ∈ L[R → R] and p ∈ domR g, we have   g (p; χY ) ≥ 0 (∀ Y ⊆ V ), g(p) ≤ g(q) (∀ q ∈ RV ) ⇐⇒ g  (p; 1) = 0. (2) For a polyhedral L -convex function g ∈ L [R → R] and p ∈ domR g, we have g(p) ≤ g(q)

(∀ q ∈ RV ) ⇐⇒ g  (p; ±χY ) ≥ 0

(∀ Y ⊆ V ).

Proposition 7.34. Let g ∈ L[R → R] be a polyhedral L-convex function. For any x ∈ RV , arg min g[−x] is an L-convex polyhedron if it is not empty. The property in Proposition 7.34 characterizes polyhedral L-convexity, to be shown in Theorem 7.45. Note 7.35. The proviso g U > −∞ in Theorem 7.31 (4) can be weakened to g U (p0 ) > −∞ for some p0 , and similarly for g˜ > −∞ in Theorem 7.31 (5). Note 7.36. For L -convex functions on integer points we have seen characterizations in terms of discrete midpoint convexity and submodular integral convexity (Theorems 7.7 and 7.21). These characterizations, however, do not carry over to polyhedral L -convex functions. In contrast, translation submodularity (SBF [Z]) is generalized to (SBF [R]), as stated in Theorem 7.28.

7.9

Positively Homogeneous L-Convex Functions

Positively homogeneous L-convex functions coincide with the Lov´ asz extensions of submodular set functions. We denote by 0 L[R → R] the set of polyhedral L-convex functions that are positively homogeneous in the sense of (3.32) and by 0 L[Z|R → R] the set of

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Chapter 7. L-Convex Functions

integral polyhedral L-convex functions that are positively homogeneous. Also we denote by 0 L[Z → R] the set of L-convex functions g ∈ L[Z → R] on integer points such that the convex extensions g are positively homogeneous. These three families of functions can be identified with each other, i.e., 0 L[Z

→ R]  0 L[Z|R → R] = 0 L[R → R],

(7.34)

by the following proposition. We introduce yet another notation, 0 L[Z → Z], for the set of integer-valued functions belonging to 0 L[Z → R]. Proposition 7.37. (1) 0 L[Z|R → R] = 0 L[R → R]. (2) The convex extension of a function in 0 L[Z → R] belongs to 0 L[R → R]. Proof. (1) Take g ∈ 0 L[R → R]. For any x ∈ RV , arg min g[−x] is a cone that is an L-convex polyhedron (or empty) by Proposition 7.34. Hence, arg min g[−x] = D(γ) for a {0, +∞}-valued distance function γ; see section 5.6. This shows the integrality of arg min g[−x], and therefore g ∈ 0 L[Z|R → R]. (2) Take g ∈ 0 L[Z → R]. Since g is integrally convex and g is positively homogeneous, g can be represented as the maximum of a finite number of linear functions. Hence g is polyhedral, and g ∈ L[R → R] by Theorem 7.26. A positively homogeneous L-convex function g induces a submodular set function ρg by (X ⊆ V ). (7.35) ρg (X) = g(χX ) More precisely, we have the following statements, where S[R] and S[Z] denote respectively the sets of real-valued and integer-valued submodular set functions defined in (4.10) and (4.11). Proposition 7.38. (1) For g ∈ 0 L[R → R], we have ρg ∈ S[R]. (2) For g ∈ 0 L[Z → Z], we have ρg ∈ S[Z]. Proof. The submodularity of ρg is immediate from that of g. Note also that ρg (∅) = g(0) = 0 and ρg (V ) = g(1) < +∞. Conversely, the Lov´asz extension ρˆ of a submodular set function ρ ∈ S[R] is a positively homogeneous L-convex function. We recall from (4.6) the definition ρˆ(p) =

m−1 

(ˆ pi − pˆi+1 )ρ(Ui ) + pˆm ρ(Um )

(p ∈ RV ),

(7.36)

i=1

where pˆ1 > pˆ2 > · · · > pˆm denote the distinct values of the components of p, and Ui = Ui (p) = {v ∈ V | p(v) ≥ pˆi } for i = 1, . . . , m. Denote by ρˆZ the restriction of ρˆ to ZV , and note that we have ρˆZ : ZV → Z ∪ {+∞} for integer-valued ρ.

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195

Proposition 7.39. (1) For ρ ∈ S[R], we have ρˆ ∈ 0 L[R → R]. (2) For ρ ∈ S[Z], we have ρˆZ ∈ 0 L[Z → Z]. Proof. (1) We have ρˆ ∈ L[R → R] by Proposition 7.25, whereas the positive homogeneity of ρˆ is obvious. (2) follows easily from (1). The next theorem shows a one-to-one correspondence between positively homogeneous L-convex functions and submodular set functions. Theorem 7.40. For 0 L = 0 L[R → R] and S = S[R], the mappings Φ : 0 L → S and Ψ : S → 0 L defined by Φ : g → ρg in (7.35),

Ψ : ρ → ρˆ in (7.36)

are inverse to each other, establishing a one-to-one correspondence between 0 L and S. The same statement is true for 0 L = 0 L[Z → Z] and S = S[Z] with Φ : g → ρg and Ψ : ρ → ρˆZ . Proof. For ρ ∈ S we have Ψ(ρ) ∈ 0 L by Proposition 7.39 and Φ ◦ Ψ(ρ) = ρ by ρ(X) = ρˆ(χX ) in (4.7). For g ∈ 0 L we have ρ = Φ(g) ∈ S by Proposition 7.38. Denote by gZ the restriction of g to ZV . By Theorem 7.19 (5) we have g(q) = gZ (q) =

m−1 

(ˆ qi − qˆi+1 )g(χUi ) + qˆm g(χUm ) = ρˆ(q)

i=1

for q ∈ N0 , which remains valid for all q ∈ RV since g is positively homogeneous and the origin 0 is an interior point of N0 . Hence g = Ψ ◦ Φ(g). The above argument leads to the following proposition, to be used in section 7.10. Proposition 7.41. For a positively homogeneous polyhedral convex function g : RV → R ∪ {+∞} with domR g = ∅, conditions (a) and (b) below are equivalent. (a) g ∈ 0 L[R → R]. (b) arg min g[−x] ∈ L0 [R] for every x ∈ RV with inf g[−x] > −∞. Proof. (a) ⇒ (b) is immediate from Proposition 7.34. For (b) ⇒ (a), define ρ by ρ(X) = g(χX ) (X ⊆ V ) and denote by ρˆ its Lov´asz extension (7.36). Claim 1: g(p) = ρˆ(p) for all p ∈ RV . (Proof of Claim 1) The positively homogeneous convexity of g as well as (7.36) yields ρˆ(p) ≥ g(p). We may assume p ∈ domR g, since otherwise ρˆ(p) = g(p) = +∞. Take x such that p ∈ arg min g[−x], put D = arg min g[−x], and let δD be its indicator function. Since D is an L-convex cone, we have δD ∈ 0 L[R → R]. A set asz function μ defined by μ(X) = δD (χX ) (X ⊆ V ) belongs to S[R] and its Lov´ extension coincides with δD by Theorem 7.40. From this and (4.8) we see p ∈ D ⇐⇒ Ui ∈ dom μ (i = 1, . . . , m) ⇐⇒ χUi ∈ D (i = 1, . . . , m).

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In view of the expression (4.6) and the linearity of g on D we obtain g(p) = ρˆ(p). By Claim 1 and the convexity of g, ρ is submodular by Theorem 4.16. In addition we have ρ(∅) = g(0) = 0 and ρ(V ) = ρˆ(1) = g(1) < +∞. Therefore, ρ ∈ S[R]. This implies ρˆ ∈ 0 L[R → R] by Proposition 7.39 (1).

7.10

Directional Derivatives and Subgradients

Directional derivatives and subgradients of L-convex functions are considered in this section. For a polyhedral L-convex function g, the directional derivative g  (p; d) is a positively homogeneous L-convex function in d and the subgradients of g at a point form an M-convex polyhedron. Furthermore, each of these properties characterizes L-convexity. We start with directional derivatives of a polyhedral L-convex function g ∈ L[R → R]. Recall from (3.25) that, for p ∈ domR g, there exists ε > 0 such that g(p + d) − g(p) = g  (p; d)

(||d||∞ ≤ ε).

(7.37)

Proposition 7.42. If g ∈ L[R → R] and p ∈ domR g, then g  (p; ·) ∈ 0 L[R → R]. Proof. By (7.37), g  (p; ·) satisfies (SBF[R]) and (TRF[R]) in the neighborhood of d = 0. Then the claim follows from the positive homogeneity of g  (p; ·). Directional derivatives and subdifferentials of L-convex functions are given as follows. It is recalled that M0 [R], M0 [Z|R], M0 [Z], and L[R → R|Z] denote, respectively, the sets of M-convex polyhedra, integral M-convex polyhedra, M-convex sets, and dual-integral polyhedral L-convex functions. See (3.23), (6.86), and (6.88) for the notation ∂R and ∂Z , and note that g (p; ·) in Theorem 7.43 (2) denotes the directional derivative of the convex extension g of g at p. Theorem 7.43. (1) For g ∈ L[R → R] and p ∈ domR g, define ρg,p (X) = g  (p; χX ) (X ⊆ V ). Then ρg,p ∈ S[R], ∂R g(p) = B(ρg,p ) ∈ M0 [R], g  (p; ·) = ρˆg,p (·), and ∂R g(p) = ∅ in particular. If g ∈ L[R → R|Z], then ρg,p ∈ S[Z],

∂R g(p) ∈ M0 [Z|R].

(2) For g ∈ L[Z → R] and p ∈ domZ g, define ρg,p (X) = g(p + χX ) − g(p) (X ⊆ V ). Then ρg,p ∈ S[R],

∂R g(p) = B(ρg,p ) ∈ M0 [R],

g  (p; ·) = ρˆg,p (·),

and ∂R g(p) = ∅ in particular. If g ∈ L[Z → Z], then ρg,p ∈ S[Z],

∂R g(p) ∈ M0 [Z|R],

∂Z g(p) ∈ M0 [Z],

∂R g(p) = ∂Z g(p),

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and ∂Z g(p) = ∅ in particular. Proof. (1) Proposition 7.42 shows g  (p; ·) ∈ 0 L[R → R], from which follows ρg,p ∈ S[R], by Proposition 7.38. By Theorem 7.33 (1) (L-optimality criterion), x ∈ ∂R g(p) ⇐⇒ g(p + q) − g(p) ≥ q, x

(∀ q ∈ RV ) ⇐⇒ g  (p; χX ) ≥ x(X) (X ⊆ V ), g  (p; 1) = x(V ) ⇐⇒ x ∈ B(ρg,p ). We have B(ρg,p ) ∈ M0 [R] by (4.39) and g  (p; ·) = ρˆg,p (·) by (3.31), (3.33), and (4.14). If g ∈ L[R → R|Z], ∂R g(p) is an integral polyhedron by (6.76) and ρg,p (X) = sup{x(X) | x ∈ ∂R g(p)} ∈ Z. (2) It is easy to see ρg,p ∈ S[R]. The rest of the proof is similar to (1), where we use Theorem 7.14 (1) instead of Theorem 7.33 (1) and Theorem 4.15 instead of (4.39). We have consistency between (1) and (2) in Theorem 7.43. Proposition 7.44. For g ∈ L[Z|R → R] and p ∈ domR g∩ZV , we have g  (p; χX ) = g(p + χX ) − g(p) for X ⊆ V . Proof. This follows from integrality (6.75) and (5.19). The next theorem affords characterizations of polyhedral L-convex functions in terms of the L-convexity of directional derivatives, the M-convexity of subdifferentials, and the L-convexity of minimizers. Theorem 7.45. For a polyhedral convex function g : RV → R ∪ {+∞} with domR g = ∅, the four conditions (a), (b), (c), and (d) below are equivalent. (a) g ∈ L[R → R]. (b) g  (p; ·) ∈ 0 L[R → R] for every p ∈ domR g. (c) ∂R g(p) ∈ M0 [R] for every p ∈ domR g. (d) arg min g[−x] ∈ L0 [R] for every x ∈ RV with inf g[−x] > −∞. Proof. (a) ⇒ (b) is by Proposition 7.42, (a) ⇒ (c) by Theorem 7.43, and (a) ⇒ (d) by Proposition 7.34. (b) ⇒ (a): By (7.37), g is submodular and linear in the direction of 1 in a neighborhood of each p ∈ RV . This implies (SBF[R]) and (TRF[R]) (see Proposition 7.23). (b) ⇔ (c): This follows from the relation (δ∂R g(p) )• = g  (p; ·) in (3.33) and the one-to-one correspondence between M0 [R] and 0 L[R → R], which is a consequence of (4.39) and Theorem 7.40. (d) ⇒ (b): To use Proposition 7.41 let x ∈ RV be such that inf g  (p; ·)[−x] > −∞. Then inf g[−x] > −∞ and arg min g[−x] ∈ L0 [R] by (d). By (5.18) we have arg min g[−x] = {q ∈ RV | q(v) − q(u) ≤ γ(u, v) (u, v ∈ V )}

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for some γ ∈ T [R]. Since arg min(g  (p; ·)[−x]) is a cone, we see arg min(g  (p; ·)[−x]) = {q ∈ RV | q(v) − q(u) ≤ 0 ((u, v) ∈ Ap )}, with Ap = {(u, v) | p(v) − p(u) = γ(u, v)}. This shows that arg min(g  (p; ·)[−x]) is an L-convex cone. Then (b) follows from Proposition 7.41. An integrality consideration in the equivalence of (a) and (d) in the above theorem yields a characterization of integral polyhedral L-convex functions. Theorem 7.46. For a polyhedral convex function g : RV → R ∪ {+∞} with domR g = ∅, the two conditions (a) and (d) below are equivalent. (a) g ∈ L[Z|R → R]. (d) arg min g[−x] ∈ L0 [Z|R] for every x ∈ RV with inf g[−x] > −∞. Note 7.47. We prove ⇐ of Theorem 7.17 (1). We have arg min g[−x] ∈ L0 [Z] for every x ∈ RV by the assumption. Since an L-convex set is integrally convex (Theorem 5.10), g is an integrally convex function by Theorem 3.29. By the boundedness of dom g, the convex closure g of g is a polyhedral convex function and arg min g[−x] = arg min g[−x] ∈ L0 [Z|R]. This implies g ∈ L[Z|R → R] by Theorem 7.46 and therefore g ∈ L[Z → R].

7.11

Quasi L-Convex Functions

Quasi L-convex functions are introduced as a generalization of L-convex functions. The optimality criterion and the proximity theorem survive in this generalization. To define quasi L-convexity, we relax the submodularity inequality [g(p ∧ q) − g(p)] + [g(p ∨ q) − g(q)] ≤ 0 to sign patterns of g(p ∧ q) − g(p) and g(p ∨ q) − g(q) compatible with (implied by) this inequality, which are given as follows: g(p ∧ q) − g(p) \ g(p ∨ q) − g(q) − 0 +

− # # #

0 # # ×

+ # × ×

Here # and × denote possible and impossible cases, respectively. We call a function g : ZV → R ∪ {+∞} quasi submodular if it satisfies the following: (QSB) For any p, q ∈ ZV , g(p ∧ q) ≤ g(p) or g(p ∨ q) ≤ g(q). Since p and q are symmetric, (QSB) implies also that g(p ∧ q) ≤ g(q) or g(p ∨ q) ≤ g(p). Similarly, we call g semistrictly quasi submodular if it satisfies the following property:51 51 The condition (SSQSB) was introduced by Milgrom–Shannon [129], in which g : ZV → R ∪ {−∞} is called quasi supermodular if −g satisfies (SSQSB).

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(SSQSB) For any p, q ∈ ZV , both (i) and (ii) hold: (i) g(p ∨ q) ≥ g(q) =⇒ g(p ∧ q) ≤ g(p), (ii) g(p ∧ q) ≥ g(p) =⇒ g(p ∨ q) ≤ g(q). Furthermore, a function g : ZV → R ∪ {+∞} with dom g = ∅ is called quasi L-convex if it satisfies (QSB) and (TRF[Z]) and semistrictly quasi L-convex if it satisfies (SSQSB) and (TRF[Z]). Example 7.48. A quasi L-convex function arises from a nonlinear scaling of an L-convex function. For a submodular function g : ZV → R ∪ {+∞} and a function ϕ : R → R ∪ {+∞}, define g˜ : ZV → R ∪ {+∞} by  ϕ(g(p)) (p ∈ dom g), g˜(p) = (7.38) +∞ (p ∈ / dom g). Then g˜ satisfies (QSB) if ϕ is nondecreasing and (SSQSB) if ϕ is strictly increasing. If g satisfies (TRF[Z]) with r = 0, this property is inherited by g˜. Weaker variants of (QSB) and (SSQSB) can be conceived by considering possible sign patterns of the four values g(p ∧ q) − g(p), g(p ∧ q) − g(q), g(p ∨ q) − g(p), and g(p ∨ q) − g(q). (QSBw ) For any p, q ∈ dom g, max{g(p), g(q)} ≥ min{g(p∧q), g(p∨q)}. (SSQSBw ) For any p, q ∈ dom g, either (i) or (ii) holds: (i) max{g(p), g(q)} > min{g(p ∧ q), g(p ∨ q)}, (ii) g(p) = g(q) = g(p ∧ q) = g(p ∨ q). The relationship among various versions of quasi submodularity is summarized as follows. The second statement below shows that all the conditions are equivalent for g if they are imposed on every perturbation of g by a linear function. Recall the definition of g[x]; i.e., g[x](p) = g(p) + p, x . Theorem 7.49. For g : ZV → R ∪ {+∞}, the following implications hold true. (1)

(SBF[Z])

=⇒

(SSQSB) ⇓ (SSQSBw )

=⇒ =⇒

(QSB) ⇓ (QSBw ).

(2) g satisfies (SBF[Z]) ⇐⇒ ∀ x ∈ RV , g[x] satisfies (QSBw ). Proof. (1) This is immediate from the definitions. (2) Combining Theorems 7.51 and 7.52 below establishes this. As is easily seen from the definitions of quasi L-convexity, most of the properties of quasi-submodular functions can be restated naturally in terms of quasi L-convex functions, and vice versa. We will work mainly with quasi-submodular functions. The following are quasi versions of Theorem 7.2 for L-convex functions.

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Proposition 7.50. Assume that g : ZV → R ∪ {+∞} satisfies g(p) = g(p + 1) for all p ∈ ZV . (1) For g satisfying (QSBw ) and for p, q ∈ ZV and α ∈ Z, we have max{g(p), g(q)} ≥ min{g(p ∨ (q − α1)), g((p + α1) ∧ q)}.

(7.39)

In particular, for p, q ∈ dom g and α ∈ [0, α1 − α2 ]Z , we have max{g(p), g(q)} ≥ min{g(p + αχX ), g(q − αχX )},

(7.40)

where X ⊆ V , α1 ∈ Z, and α2 ∈ Z ∪ {−∞} are defined by X = arg max{q(v) − p(v)}, v∈V

α1 = max{q(v) − p(v)}, v∈V

α2 = max {q(v) − p(v)}. v∈V \X

(2) For g satisfying (SSQSBw ) and for p, q ∈ ZV with g(p) = g(q) and α ∈ Z, we have inequality (7.39) with strict inequality. In particular, for p, q ∈ dom g with g(p) = g(q) and α ∈ [0, α1 − α2 ]Z , we have (7.40) with strict inequality. (3) For g satisfying (SSQSB) and for p, q ∈ ZV and α ∈ Z, we have g(p ∨ (q − α1)) ≥ g(p) =⇒ g((p + α1) ∧ q) ≤ g(q),

(7.41)

g((p + α1) ∧ q) ≥ g(q) =⇒ g(p ∨ (q − α1)) ≤ g(p).

(7.42)

In particular, for p, q ∈ dom g and α ∈ [0, α1 − α2 ]Z , we have g(p + αχX ) ≥ g(p) =⇒ g(q − αχX ) ≤ g(q), g(q − αχX ) ≥ g(q) =⇒ g(p + αχX ) ≤ g(p).

(7.43) (7.44)

Proof. Inequality (7.39) follows from max{g(p), g(q)} = max{g(p), g(q − α1)} ≥ min{g(p ∨ (q − α1)), g(p ∧ (q − α1))}, in which g(p ∧ (q − α1)) = g((p ∧ (q − α1)) + α1) = g((p + α1) ∧ q). Inequality (7.40) is obvious from (7.39) since p ∨ {q − (α1 − α)1} = p + αχX and (p + (α1 − α)1) ∧ q = q − αχX for α ∈ [0, α1 − α2 ]Z . The proofs of (2) and (3) are similar. The quasi submodularity of a set D ⊆ ZV can be defined as the quasi submodularity of the indicator function δD : ZV → {0, +∞}. (QSB) for δD is equivalent to (QDL) p, q ∈ D =⇒ p ∧ q ∈ D or p ∨ q ∈ D for D, whereas (SSQSB) for δD is equivalent to (SBS[Z]) for D. Level sets of quasi-submodular functions have quasi submodularity. Furthermore, the weaker version (QSBw ) of quasi submodularity for functions can be characterized by the property (QDL) of level sets; recall the notation L(g, α) from (6.95). Theorem 7.51. A function g : ZV → R ∪ {+∞} satisfies (QSBw ) if and only if the level set L(g, α) satisfies (QDL) for every α ∈ R.

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Proof. For the “if” part, take p, q ∈ dom g and put α = max{g(p), g(q)}. Since p, q ∈ L(g, α), we have p ∧ q ∈ L(g, α) or p ∨ q ∈ L(g, α); i.e., max{g(p), g(q)} ≥ min{g(p ∧ q), g(p ∨ q)}. The “only if” part is even easier. A submodular function over the integer lattice can be characterized by using level sets of functions perturbed by linear functions. Theorem 7.52. A function g : ZV → R ∪ {+∞} satisfies (SBF[Z]) if and only if the level set L(g[x], α) satisfies (QDL) for all x ∈ RV and α ∈ R. Proof. The “only if” part follows from Theorem 7.51 and the submodularity of g[x]. For the proof of the “if” part, take p, q ∈ dom g. By (QDL) for L(g, max{g(p), g(q)}) we have p ∧ q ∈ dom g or p ∨ q ∈ dom g. We consider the former case, where we may assume p ∧ q = p, q. For any ε > 0, we can choose some x ∈ RV such that g[x](p) = g[x](q) = g[x](p ∧ q) − ε. (QDL) for L(g[x], α) with α = g[x](p) shows p ∨ q ∈ L(g[x], α), which implies g[x](p) + g[x](q) = 2α ≥ g[x](p ∧ q) + g[x](p ∨ q) − ε. Since ε > 0 is arbitrary, this means (SBF[Z]). Next we turn to the minimization of a quasi L-convex function. We assume r = 0 in (TRF[Z]) since otherwise no minimizer exists. Global minimality is characterized by local minimality. Theorem 7.53 (Quasi L-optimality criterion). Assume that g : ZV → R ∪ {+∞} satisfies g(p) = g(p + 1) (∀ p ∈ ZV ). (1) For g satisfying (QSBw ) and p ∈ dom g, we have: g(p) < g(q) for all q ∈ ZV such that q − p is not a multiple of 1 ⇐⇒ g(p) < g(p + χX ) for all X ⊆ V with X ∈ / {∅, V }. (2) For g satisfying (SSQSBw ) and p ∈ dom g, we have: g(p) ≤ g(q) (∀ q ∈ ZV ) ⇐⇒ g(p) ≤ g(p + χX ) (∀ X ⊆ V ). Proof. We prove ⇐ of (1) by contradiction. Suppose that g(q) ≤ g(p) for some q ∈ dom g such that q − p is not a multiple of 1. We may assume that q ≥ p by (TRF[Z]) with r = 0 and that q minimizes maxv∈V {q(v) − p(v)} among such vectors. Put X = arg maxv∈V {q(v) − p(v)}, where X = V . By Proposition 7.50 (1), we obtain g(p) = max{g(p), g(q)} ≥ min{g(p + χX ), g(q − χX )}, whereas g(p) < g(q − χX ) by the choice of q. Hence follows g(p) ≥ g(p + χX ), a contradiction to the strict local minimality of p. The other direction ⇒ of (1) is obvious, and (2) can be shown similarly by Proposition 7.50 (2). The proximity theorem for L-convex functions (Theorem 7.18) can be generalized for quasi L-convex functions. Theorem 7.54 (Quasi L-proximity theorem). Let g : ZV → R ∪ {+∞} be a function satisfying (SSQSB) and g(p) = g(p + 1) (∀ p ∈ ZV ), and assume n = |V |

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and α ∈ Z++ . If pα ∈ dom g satisfies (7.21), then arg min g = ∅ and there exists p∗ ∈ arg min g with (7.22). Proof. The proof of Theorem 7.18 works with (7.43) and (7.44) in place of (L -APR[Z]).

Bibliographical Notes The concept of L-convex functions was introduced by Murota [140]. L -convex functions are defined by Fujishige–Murota [68] as a variant of L-convex functions, together with the observation that they coincide with the submodular integrally convex functions considered earlier by Favati–Tardella [49]. Theorems 7.1 and 7.3 are due to [68], and Theorem 7.2 is stated in Murota [147]. Discrete midpoint convexity is considered by Favati–Tardella [49] with an observation of its equivalence to submodular integral convexity. The equivalence of discrete midpoint convexity to translation submodularity (SBF [Z]) in Theorem 7.7 is by Fujishige–Murota [68], whereas that to (L -APR[Z]) is noted in Murota [147]. Condition (7.11) for quadratic L-convex functions is given in Murota [141]. Separable convex functions with chain conditions (7.13) are considered in Best– Chakravarti–Ubhaya [11]. Multimodular functions are treated in Hajek [85]. The basic operations in section 7.4 are listed in Murota [141], [144], [147]. The theorems on minimizers of L-convex functions are of fundamental importance. Theorem 7.14 (L-optimality criterion) is stated in Murota [145]. Theorem 7.15 (optimality for submodular set functions) can be found as Theorem 7.2 in Fujishige [65]. Theorem 7.17 (characterization by minimizers) is a corollary of Theorem 7.45 due to Murota–Shioura [152]. A thorough study of minimizers of submodular functions is made in Topkis [202]. The L-proximity theorem (Theorem 7.18) is due to Iwata–Shigeno [105]. The present proof based on (L -APR[Z]) is by Murota–Shioura [154]. The construction of the convex extension of an L-convex function by means of the Lov´ asz extensions (Theorem 7.19) is due to Murota [140]. The same idea, however, was used earlier by Favati–Tardella [49] for submodular integrally convex functions. The equivalence of L -convexity to submodular integral convexity (Theorem 7.21) is due to Fujishige–Murota [68]. Polyhedral L-convex functions are investigated by Murota–Shioura [152], to which all the theorems in section 7.8 (Theorems 7.26, 7.28, 7.29, 7.30, 7.31, 7.32, and 7.33) as well as Proposition 7.34 are ascribed. L-convexity for nonpolyhedral convex functions is considered in Murota–Shioura [156], [157]. The correspondence between positively homogeneous L-convex functions and submodular set functions (Theorem 7.40) is established for the case of Z in Murota [140] and generalized to the case of R in Murota–Shioura [152]. Proposition 7.37 is stated in Murota [147]. Theorem 7.43 for directional derivatives and subgradients is shown for the case of Z in Murota [140], [141], and generalized to the case of R in Murota–Shioura [152]. Theorem 7.45 (characterizations in terms of directional derivatives, subdifferentials,

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and minimizers) is by [152], whereas its ramification with integrality (Theorem 7.46) is stated in Murota [147]. The concept of quasi L-convex functions was introduced by Murota–Shioura [154] on the basis of the idea of Milgrom–Shannon [129]. Theorem 7.52 is due to [129], and the other theorems in section 7.11 (Theorems 7.49, 7.51, 7.53, and 7.54) are in [154].

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Chapter 8

Conjugacy and Duality

By addressing the issues of conjugacy and duality, this chapter provides the theoretical climax of discrete convex analysis. Whereas conjugacy in convex analysis gives a symmetric one-to-one correspondence within a single class of closed convex functions, conjugacy in discrete convex analysis establishes a one-to-one correspondence between two different classes of discrete functions with different combinatorial properties distinguished by “L” and “M.” The conjugacy between L-convexity and M-convexity is thus one of the most remarkable features of discrete convex analysis. Discrete duality is another distinguishing feature. It is expressed in a number of theorems, such as the separation theorems for M-convex/M-concave functions and for L-convex/L-concave functions (M- and L-separation theorems) and the Fenchel-type duality theorem. Besides formal parallelism with convex analysis, these discrete duality theorems carry deep combinatorial facts, implying, for example, Edmonds’s intersection theorem and Frank’s discrete separation theorem for submodular set functions as special cases.

8.1

Conjugacy

M-convex functions and L-convex functions form two distinct classes of discrete functions that are conjugate to each other under the Legendre–Fenchel transformation. This stands in sharp contrast with conjugacy in convex analysis, which is a symmetric one-to-one correspondence within a single class of closed convex functions. The conjugacy correspondence between M-convexity and L-convexity is in fact a translation of two different combinatorial properties, exchangeability and submodularity, on top of convexity. The relationship between submodularity and supermodularity with respect to conjugacy is discussed first in section 8.1.1. Conjugacy for polyhedral M-/L-convex functions is established in section 8.1.2 and that for integer-valued M-/L-convex functions on integer points in section 8.1.3. 205

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8.1.1

Chapter 8. Conjugacy and Duality

Submodularity under Conjugacy

Submodularity and supermodularity are not symmetric under the Legendre–Fenchel transformation. The conjugate of a submodular function is always supermodular, whereas the conjugate of a supermodular function is not necessarily submodular. In this subsection we assume that f is a function in real variables, f : RV → R ∪ {+∞}, with a nonempty effective domain. Recall that f is submodular if f (x) + f (y) ≥ f (x ∨ y) + f (x ∧ y)

(x, y ∈ RV )

(8.1)

(x, y ∈ RV ).

(8.2)

and supermodular if f (x) + f (y) ≤ f (x ∨ y) + f (x ∧ y)

Also recall from (3.26) that the Legendre–Fenchel transform f • : RV → R ∪ {+∞} is defined by f • (p) = sup{ p, x − f (x) | x ∈ RV }

(p ∈ RV ).

(8.3)

Theorem 8.1. For a submodular function f , the Legendre–Fenchel transform f • is supermodular. Proof. For x, y ∈ RV and p, q ∈ RV , we have p, x + q, y ≤ p ∨ q, x ∨ y + p ∧ q, x ∧ y . From this inequality, submodularity (8.1), and the definition (8.3), we see that [ p, x − f (x)] + [ q, y − f (y)] ≤ [ p ∨ q, x ∨ y − f (x ∨ y)] + [ p ∧ q, x ∧ y − f (x ∧ y)] ≤ f • (p ∨ q) + f • (p ∧ q). Taking the supremum over x and y we obtain f • (p) + f • (q) ≤ f • (p ∨ q) + f • (p ∧ q), which shows the supermodularity of f • . In contrast to Theorem 8.1, the Legendre–Fenchel transform of a supermodular function is not necessarily submodular. For example, consider a pair of convex quadratic functions f (x) = 12 x Ax and g(p) = 12 p A−1 p with ⎡ ⎡ ⎤ ⎤ 8 4 1 48 −28 8 1⎣ 1 ⎣ −28 A= 4 8 4 ⎦ , A−1 = 63 −28 ⎦ . 8 35 1 4 8 8 −28 48 We have g = f • by Proposition 2.9, whereas f is supermodular and g is not submodular by Proposition 2.6.

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If n = 2, however, supermodularity does imply submodularity of the Legendre– Fenchel transform. Proposition 8.2. For a supermodular function f in two variables, the Legendre– Fenchel transform f • is submodular. Proof. It suffices to show that f • (p(1), p(2)) + f • (q(1), q(2)) ≤ f • (p(1), q(2)) + f • (q(1), p(2))

(8.4)

for p = (p(1), p(2)) ∈ R2 and q = (q(1), q(2)) ∈ R2 with p(1) ≥ q(1) and p(2) ≥ q(2). We claim that [ p, x − f (x)] + [ q, y − f (y)] ≤ f • (p(1), q(2)) + f • (q(1), p(2))

(8.5)

for any x = (x(1), x(2)) ∈ R2 and y = (y(1), y(2)) ∈ R2 . The inequality (8.4) is an immediate consequence of (8.5), since the supremum of the left-hand side of (8.5) over x and y coincides with the left-hand side of (8.4). Proof of (8.5): If x(1) ≥ y(1) and x(2) ≥ y(2), we have p, x + q, y = (p(1), q(2)), (x(1), y(2)) + (q(1), p(2)), (y(1), x(2)) , f (x) + f (y) ≥ f (x(1), y(2)) + f (y(1), x(2)), and, therefore, LHS of (8.5) ≤ [ (p(1), q(2)), (x(1), y(2)) − f (x(1), y(2))] + [ (q(1), p(2)), (y(1), x(2)) − f (y(1), x(2))] ≤ f • (p(1), q(2)) + f • (q(1), p(2)) = RHS of (8.5). If x(1) ≤ y(1), we have p, x + q, y ≤ (p(1), q(2)), y + (q(1), p(2)), x , and, therefore, LHS of (8.5) ≤ [ (p(1), q(2)), y − f (y)] + [ (q(1), p(2)), x − f (x)] ≤ f • (p(1), q(2)) + f • (q(1), p(2)) = RHS of (8.5). A similar argument holds for the case of x(2) ≤ y(2). By Theorem 8.1, submodularity is preserved under the transformation f → −f • . However, this does not establish a symmetric one-to-one correspondence within the class of submodular functions. It is not true, either, that the mapping f → f • gives a one-to-one correspondence between the class of submodular functions and the class of supermodular functions.

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Polyhedral M-/L-Convex Functions

Conjugacy for polyhedral M-convex and L-convex functions is considered here. We start with a technical lemma. Proposition 8.3. Let g ∈ L[R → R] be a polyhedral L-convex function. For x, y ∈ RV with inf g[−x] > −∞ and inf g[−y] > −∞ and for u ∈ supp+ (x − y), there exists v ∈ supp− (x − y) such that p(v) − p(u) ≤ q(v) − q(u)

(∀ p ∈ arg min g[−x], ∀ q ∈ arg min g[−y]).

(8.6)

Proof. We may assume arg min g[−x] = ∅ and arg min g[−y] = ∅. By Proposition 7.34, we have arg min g[−x] ∈ L0 [R] and arg min g[−y] ∈ L0 [R]. It suffices to demonstrate the existence of v ∈ supp− (x − y) such that p(v) ≤ q(v) for all p ∈ Dx and q ∈ Dy , where Dx = {p | p ∈ arg min g[−x], p(u) = 0} ∈ L0 [R], Dy = {q | q ∈ arg min g[−y], q(u) = 0} ∈ L0 [R]. To prove this by contradiction, suppose that for every v ∈ supp− (x − y) there exist pv ∈ Dx and qv ∈ Dy with pv (v) > qv (v). Then, for ; < p∗ = {pv | v ∈ supp− (x − y)}, q∗ = {qv | v ∈ supp− (x − y)}, we have p∗ ∈ Dx , q∗ ∈ Dy , and p∗ (v) > q∗ (v)  p∗ (v) − λ  p = (p∗ − λ1) ∨ q∗ =  q∗ (v) q∗ (v) + λ q  = p∗ ∧ (q∗ + λ1) = p∗ (v)

(∀ v ∈ supp− (x − y)). By defining (v ∈ supp+ (p∗ − q∗ )), (v ∈ V \ supp+ (p∗ − q∗ )), (v ∈ supp+ (p∗ − q∗ )), (v ∈ V \ supp+ (p∗ − q∗ )),

with λ = min{p∗ (v) − q∗ (v) | v ∈ supp+ (p∗ − q∗ )} > 0, we obtain g(p∗ ) + g(q∗ ) ≥ g(p ) + g(q  )

(8.7)

from Theorem 7.29. By supp− (x − y) ⊆ supp+ (p∗ − q∗ ), on the other hand, we see p , x + q  , y − p∗ , x − q∗ , y

 =λ {y(v) − x(v) | v ∈ supp+ (p∗ − q∗ )} = > (q∗ (v) − p∗ (v))(x(v) − y(v)) | v ∈ V \ supp+ (p∗ − q∗ ) +  ≥λ {y(v) − x(v) | v ∈ V \ {u}} = λ{x(u) − y(u)} > 0,

(8.8)

where the last equality is due to x(V ) = y(V ) = r (the constant in (TRF[R])). Combining (8.7) and (8.8) results in g[−x](p ) + g[−y](q  ) < g[−x](p∗ ) + g[−y](q∗ ),

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209

which is a contradiction to p∗ ∈ arg min g[−x] and q∗ ∈ arg min g[−y]. The conjugacy theorem for polyhedral M-convex and L-convex functions is now stated. Theorem 8.4 (Conjugacy theorem). (1) The classes of polyhedral M-convex functions and polyhedral L-convex functions, M = M[R → R] and L = L[R → R], are in one-to-one correspondence under the Legendre–Fenchel transformation (8.3). That is, for f ∈ M and g ∈ L, we have f • ∈ L, g • ∈ M, f •• = f , and g •• = g. (2) The classes of polyhedral M -convex functions and polyhedral L -convex functions, M [R → R] and L [R → R], are in one-to-one correspondence under (8.3) in a similar manner. Proof. (1) and (2) are equivalent, so we prove (1). We first note that f •• = f for any polyhedral convex function f . For f ∈ M we have ∂R f • (p) = arg min f [−p] ∈ M0 [R] (∀ p ∈ dom f • ) by Proposition 6.53. Then f • ∈ L by (c)⇒(a) in Theorem 7.45. (An alternative proof is described in the proof of Theorem 8.6.) Conversely, take g ∈ L and x, y ∈ dom g • . Since inf g[−x] > −∞ and inf g[−y] > −∞, Proposition 8.3 shows that for every u ∈ supp+ (x − y) there exists v ∈ supp− (x − y) satisfying (8.6). Noting that (δarg min g[−x] )• = (g • ) (x; ·), which follows from (3.30) and (3.33), we obtain (g • ) (x; v, u) + (g • ) (y; u, v) = sup{p(v) − p(u) | p ∈ arg min g[−x]} + sup{q(u) − q(v) | q ∈ arg min g[−y]} ≤ 0. This shows (M-EXC [R]) for g • , and hence g • ∈ M. Theorem 8.4 (2) states that L -convex functions and M -convex functions are transformed to each other, where L -convex functions are submodular (Theorem 7.28) and M -convex functions are supermodular (Theorem 6.51). It is noted that Theorem 8.4 (2) does not imply, nor is it implied by, Theorem 8.1, which shows the supermodularity of the conjugate of a submodular function. Recalling the basic fact that the conjugate of the indicator function of a convex set is a positively homogeneous convex function, and vice versa, we see from Theorem 8.4 above that M-convex polyhedra M0 [R] and positively homogeneous L-convex functions 0 L[R → R] are conjugate to each other and also that L-convex polyhedra L0 [R] and positively homogeneous M-convex functions 0 M[R → R] are conjugate to each other. On the other hand, we can identify positively homogeneous L-convex functions 0 L[R → R] with submodular set functions S[R] (Theorem 7.40) and positively homogeneous M-convex functions 0 M[R → R] with distance functions with the triangle inequality T [R] (Theorem 6.59). We can summarize these

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one-to-one correspondences in the following diagram: M0 [R] ←→ M[R → R] ←→ T [R] ←→ 0 M[R → R] ←→

0 L[R

→ R] ←→ S[R] L[R → R] L0 [R]

(8.9)

In addition, the polarity between M-convex cones and L-convex cones follows from (8.9). This is because two convex cones are polar to each other if and only if their indicator functions are conjugate to each other. Thus we obtain the following theorem. Theorem 8.5. A polyhedral cone is M-convex if and only if its polar cone is Lconvex. Hence, the classes of M-convex cones and L-convex cones are in one-to-one correspondence under polarity (3.34). Taking integrality into account in diagram (8.9), we obtain

T [Z] ←→

M0 [Z|R] M[Z|R → R] M[R → R|Z] 0 M[R → R|Z]

←→ ←→ ←→ ←→

0 L[R

→ R|Z] ←→ S[Z] L[R → R|Z] L[Z|R → R] L0 [Z|R]

(8.10)

where M[Z|R → R] and L[Z|R → R] denote the sets of integral polyhedral Mconvex and L-convex functions, respectively; M[R → R|Z] and L[R → R|Z] denote the sets of dual-integral polyhedral M-convex and L-convex functions, respectively; and 0 M[R → R|Z] and 0 L[R → R|Z] are their subclasses with positive homogeneity.52 It is known that the conjugacy relationship between M-convexity and Lconvexity holds more generally for closed proper convex functions. Recall that the Legendre–Fenchel transformation gives a symmetric one-to-one correspondence in the class of all closed proper convex functions (Theorem 3.2). Theorem 8.6. A closed proper convex function f satisfies (M-EXC[R]) if and only if f = g • for a closed proper convex function g that satisfies (SBF[R]) and (TRF[R]). Proof. The proof of the “if” part is essentially the same as the latter half of the proof of Theorem 8.4. The “only if” part needs a new approach, since the implication (c)⇒(a) in Theorem 7.45 does not carry over to nonpolyhedral convex functions. Suppose that f satisfies (M-EXC[R]) and put g = f • . It is easy to show (TRF[R]) for g. The proof of the submodularity (SBF[R]) for g consists of the following steps. 52 The notation for dual integrality extends naturally to other classes of functions. For example, M [R → R|Z] and L [R → R|Z] denote the sets of dual-integral polyhedral M -convex and L -convex functions, respectively.

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1. We may assume that dom f is bounded, so that dom g = RV . 2. For p0 ∈ RV and U ⊆ V with |U | = 2, denote by fˆ : RU → R ∪ {+∞} the projection of f [−p0 ] to U and by gˆ : RU → R the restriction of g(p0 + p) to U . Then we have gˆ = (fˆ)• . 3. (M-EXC[R]) of f implies the supermodularity of fˆ. 4. The supermodularity of fˆ implies the submodularity of (fˆ)• by Proposition 8.2. 5. The submodularity of gˆ for any p0 and any U implies the submodularity of g. The details are given in Murota–Shioura [156], [157].

Note 8.7. With Theorem 8.4 we complete the proof of Theorem 6.63 (characterizations of polyhedral M-convex functions). (b) ⇔ (c) follows from δ∂R f (x) • = f  (x; ·) in (3.33) and the correspondence between L0 [R] and 0 M[R → R], which is a special case of Theorem 8.4. To show (a) ⇔ (c) ⇔ (d), put g = f • and note that ∂R f (x) = arg min g[−x] and arg min f [−p] = ∂R g(p). By Theorem 8.4 and Theorem 7.45 (characterizations of polyhedral L-convex functions), we see that f ∈ M[R → R] ⇔ g ∈ L[R → R] ⇔ arg min g[−x] ∈ L0 [R] ⇔ ∂R g(p) ∈ M0 [R]. Note 8.8. We complete the proof of Theorem 6.45 using Theorem 6.63 ((d) ⇒ (a)) established in Note 8.7. Let f be the convex extension of f ∈ M[Z → R]. For any p ∈ RV , arg min f [−p] is an M-convex polyhedron if it is not empty (Theorem 6.43). Since f is polyhedral by the assumption, Theorem 6.63 ((d) ⇒ (a)) shows f ∈ M[R → R]. Note 8.9. Using Theorem 8.5, we complete the proof of (4.42), the representation of an M-convex cone in terms of vectors χu − χv (u, v ∈ V ). Let B be an Mconvex cone and D be the polar of B. By Theorem 8.5, D is an L-convex cone, and by (5.18) it can be represented as D = {p ∈ RV | p, ai ≤ 0 (i = 1, . . . , m)} for some ai = χui − χvi (i = 1, . . . , m). Since B is the polar of D, this implies B = {x ∈ RV | x is a nonnegative combination of ai (i = 1, . . . , m)} by (3.36). Conversely, a convex cone of this form is M-convex, as can be shown by reversing the above argument. Note 8.10. Using Theorem 8.5, we complete the proof of (5.21), the representation of an L-convex cone in terms of a ring family D ⊆ 2V . Let D be an L-convex cone and B be the polar of D. By Theorem 8.5, B is an M-convex cone, and, by (4.39), it can be represented as B = B(ρ) using ρ ∈ S[R] with ρ : 2V → {0, +∞}; i.e., B = {x ∈ RV | χX , x ≤ 0 (∀ X ∈ D \ {V }), χV , x = 0} with D = dom ρ, which is a ring family. Since D is the polar of B, this implies ) ( * )  ) D= p= cX χX ) cX ≥ 0 (X ∈ D \ {V }) ) X∈D

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by (3.36). Conversely, a convex cone of this form is L-convex, as can be shown by reversing the above argument.

8.1.3

Integral M-/L-Convex Functions

We turn to functions defined on integer points. For functions f : ZV → R ∪ {+∞} and h : ZV → R ∪ {−∞}, discrete versions of the Legendre–Fenchel transformations are defined by f • (p) = sup{ p, x − f (x) | x ∈ ZV } ◦

h (p) = inf{ p, x − h(x) | x ∈ Z } V

(p ∈ RV ), (p ∈ R ). V

(8.11) (8.12)

We call (8.11) and (8.12), respectively, convex and concave discrete Legendre– Fenchel transformations. The functions f • : RV → R ∪ {±∞} and h◦ : RV → R ∪ {±∞} are called the convex conjugate of f and the concave conjugate of h, respectively. Note that h◦ (p) = −(−h)• (−p). For an integer-valued function f , f • (p) is integral for an integer vector p. Hence, (8.11) with p ∈ ZV defines a transformation of f : ZV → Z ∪ {+∞} to f • : ZV → Z ∪ {±∞}; we refer to (8.11) with p ∈ ZV as (8.11)Z . We call (f • )• using (8.11)Z the integer biconjugate of f and denote it by f •• . Similarly, (8.12) with p ∈ ZV is designated by (8.12)Z and we define h◦◦ = (h◦ )◦ . The following fact is fundamental for the conjugacy of discrete functions. Proposition 8.11. For a function f : ZV → Z ∪ {+∞} and a point x ∈ domZ f , we have f •• (x) = f (x) if ∂Z f (x) = ∅, where f •• means the integer biconjugate with respect to the discrete Legendre–Fenchel transformation (8.11)Z . Proof. For p ∈ ∂Z f (x), we have f • (p) = p, x − f (x) (cf. (3.30)), and therefore f •• (x) = sup{ q, x − f • (q) | q ∈ ZV } ≥ p, x − f • (p) = f (x). On the other hand, f •• (x) ≤ f (x) for any f and x. The conjugacy theorem for discrete M-convex and L-convex functions reads as follows. Theorem 8.12 (Discrete conjugacy theorem). (1) The classes of integer-valued M-convex functions and integer-valued Lconvex functions, M = M[Z → Z] and L = L[Z → Z], are in one-to-one correspondence under the discrete Legendre–Fenchel transformation (8.11)Z . That is, for f ∈ M and g ∈ L, we have f • ∈ L, g • ∈ M, f •• = f , and g •• = g. (2) The classes of integer-valued M -convex functions and integer-valued L convex functions, M [Z → Z] and L [Z → Z], are in one-to-one correspondence under (8.11)Z in a similar manner. Proof. The basic idea of the proof is to apply Theorem 8.4 to the convex extensions of f and g with additional arguments for discreteness. Since (1) and (2) are equivalent, we deal with (2).

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213 •

(i) Take f ∈ M [Z → Z]. Let f be the convex extension of f and f be the • conjugate of f in the sense of (8.3). We have f • (p) = f (p) for p ∈ ZV . If domZ f is bounded, f is polyhedral convex, and therefore f ∈ M [R → R] • • by Theorem 6.45. Then Theorem 8.4 shows f ∈ L [R → R]. Since f • (p) = f (p) • for p ∈ ZV , (SBF [R]) for f implies (SBF [Z]) for f • . Hence f • ∈ L [Z → Z]. If domZ f is unbounded, we consider the restriction fk of f to integer interval [−k1, k1]Z for k ∈ Z large enough to ensure domZ f ∩ [−k1, k1]Z = ∅. Then we have fk ∈ M [Z → Z] and fk• ∈ L [Z → Z] by the argument above. For each p ∈ domZ f • , there exists kp such that f • (p) = fk• (p) for all k ≥ kp (cf. Theorem 6.42 and Proposition 3.30). Therefore, (SBF [Z]) for fk• implies (SBF [Z]) for f • . Hence f • ∈ L [Z → Z]. (ii) Take g ∈ L [Z → Z]. Let g be the convex extension of g and g • be the conjugate of g in the sense of (8.3). We have g • (x) = sup{ p, x − g(p) | p ∈ ZV }

(x ∈ RV )

(8.13)

and, in particular, g • (x) = g • (x) (x ∈ ZV ). If domZ g is bounded, g is polyhedral convex, and therefore g ∈ L [R → R] by Theorem 7.26. Then g • ∈ M [R → R] by Theorem 8.4, and g • satisfies (M EXC[R]) by Theorem 6.47. We claim that α0 = 1 is valid in (M -EXC[R]) for x, y ∈ domR g• ∩ ZV = domZ g • . Then it follows that g • satisfies (M -EXC[Z]) and g • ∈ M [Z → Z] by Theorem 6.2. To show α0 = 1, fix x, y ∈ domZ g • , u ∈ supp+ (x − y), and v ∈ supp− (x − y) ∪ {0} in (M -EXC[R]). By the assumed boundedness of domZ g, the supremum in (8.13) is attained by some p. Moreover, there exist p0 ∈ ZV and α1 > 0 such that g• (x − α(χu − χv )) = p0 , x − α(χu − χv ) − g(p0 )

(8.14)

for all α ∈ [0, α1 ]R . Condition (8.14) can be written as p0 ∈ arg min g[−x + α(χu − χv )],

(8.15)

which is equivalent, by the L-optimality criterion (Theorem 7.14 (2)), to α εχY , χv − χu ≤ g(p0 + εχY ) − g(p0 ) − εχY , x

(∀ Y ⊆ V, ε = ±1).

Note that the right-hand side is an integer and the coefficient of α on the left is either ±1 or 0. By virtue of this integrality, the above inequality is satisfied by all α ∈ [0, 1]R if it is satisfied by some α > 0. Therefore, (8.14) holds for all α ∈ [0, 1]R . Similarly, there exists q0 ∈ ZV such that g • (y + α(χu − χv )) = q0 , y + α(χu − χv ) − g(q0 ) for all α ∈ [0, 1]R . Combining (8.14) and (8.16) shows g • (x − α(χu − χv )) + g • (y + α(χu − χv )) − g • (x) − g• (y) = α[p0 (v) − p0 (u) + q0 (u) − q0 (v)]

(8.16)

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for all α ∈ [0, 1]R . Hence α0 = 1 is valid.53 If domZ g is unbounded, we consider the restriction gk of g to integer interval [−k1, k1]Z for k ∈ Z large enough to ensure domZ g ∩ [−k1, k1]Z = ∅. Then we have gk ∈ L [Z → Z] and gk• ∈ M [Z → Z] by the argument above. For each x ∈ domZ g • , there exists kx such that g • (x) = gk• (x) for all k ≥ kx (cf. Theorem 7.20 and Proposition 3.30). Therefore, (M -EXC[Z]) for gk• implies (M -EXC[Z]) for g • . Hence g • ∈ M [Z → Z]. (iii) Finally, f •• = f and g •• = g follow from Proposition 8.11, Theorem 6.61 (2), and Theorem 7.43 (2). As the discrete counterpart of diagram (8.9), we obtain the following:

T [Z] ←→

M0 [Z] M[Z → Z] 0 M[Z → Z]

←→ ←→ ←→

0 L[Z

→ Z] ←→ S[Z] L[Z → Z] L0 [Z]

(8.17)

This follows from the discrete conjugacy theorem (Theorem 8.12) in combination with Theorems 7.40, 6.59, 4.15, and 5.5. In addition, we can obtain the M -/L version of (8.17). The conjugacy relationship among discrete convex functions is schematized in Fig. 8.1, where M2 -convex and L2 -convex functions are defined in section 8.3. This is the ultimate picture for the discrete conjugacy relationship, which originated in the equivalence between the base family and the rank function of a matroid (section 2.4). In other words, the exchange property and submodularity are conjugate to each other at various levels. Examples of mutually conjugate M-convex and L-convex functions are demonstrated below for integer-valued functions defined on integer points. • In the network flow problem in section 2.2, if fa ∈ C[Z → Z] and ga ∈ C[Z → Z] are conjugate for each arc a ∈ A, then ) * ( )  ) fa (ξ(a))) ∂ξ(v) = −x(v)(v ∈ T ), ∂ξ(v) = 0(v ∈ V \ T ) , f (x) = inf ) ξ a∈A ) ( * )  ) g(p) = inf ga (η(a))) η(a) = −δ p˜(a) (a ∈ A), p˜(v) = p(v)(v ∈ T ) η,p˜ ) a∈A

in (2.42) and (2.43) are conjugate to each other. We will dwell on the conjugacy in network flow in section 9.6. • In a valuated matroid (V, B, ω), which arises, e.g., from a polynomial matrix (section 2.4.2),  −ω(J) (x = χJ , J ∈ B), f (x) = +∞ (otherwise), 53 An alternative proof is possible on the basis of (i)–(iii) below if (iii) is accepted as a known fact: (i) (8.15) is equivalent to x − α(χu − χv ) ∈ ∂R g(p0 ), (ii) ∂R g(p0 ) is an integral M -convex polyhedron (Theorem 7.43 (2)), and (iii) for an integral M -convex polyhedron Q and x, y ∈ Q∩ZV , α0 = 1 is valid in (B -EXC[R]).

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215



M2 -FNC

-

M-SET  = base polyhedron 

M-FNC

L2 -FNC

- L-PHF  submod. set fnc. conjugate

M-PHF   distance fnc.

-

L-FNC L-SET

projection ?

restriction ?

M -SET  = g-polymatroid M -FNC M -PHF

M2 -FNC



conjugate

L -PHF  strong pair

-



-



-

L -FNC L -SET

L2 -FNC

(FNC = function, SET = set, PHF = positively homogeneous function) Figure 8.1. Conjugacy in discrete convex functions. ⎫ ⎧ ) ) ⎬ ⎨  ) g(p) = max ω(J) + p(j))) J ∈ B ⎭ ⎩ ) j∈J

in (2.77) and (2.78) are conjugate to each other. • If f ∈ M [Z → Z] and g ∈ L [Z → Z] are conjugate, then the restriction fU and the projection g U to a subset U ⊆ V are conjugate to each other and the projection f U and the restriction gU are conjugate to each other. • If f ∈ M[Z → Z] and g ∈ L[Z → Z] are conjugate and ϕv ∈ C[Z → Z] and ψv ∈ C[Z → Z] are conjugate for each v ∈ V , then  f˜(x) = f (x) + ϕv (x(v)), v∈V

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216

Chapter 8. Conjugacy and Duality + g˜(p) = inf

q∈ZV

g(q) +



, ψv (p(v) − q(v))

v∈V

in (6.46) and (7.18) are conjugate to each other. • If fi ∈ M [Z → Z] and gi ∈ L [Z → Z] are conjugate for i = 1, 2, then f1 2Z f2 ∈ M [Z → Z] and g1 + g2 ∈ L [Z → Z] are conjugate to each other. Note 8.13. In section 2.1 we saw the conjugacy between M -convex and L -convex quadratic functions in real variables (Theorems 2.11 and 2.16). This conjugacy relationship does not have a discrete counterpart. Let f : Zn → Z be a quadratic function represented as f (x) = x Ax, with a positive-definite symmetric matrix A with integer entries, and f • : Zn → Z be the discrete Legendre–Fenchel transform (8.11) of f . If A satisfies (6.26) through (6.28), then f is M -convex and f • is L -convex, but f • is not necessarily a quadratic function. Likewise, if A satisfies (7.10), then f is L -convex and f • is M -convex, but f • is not necessarily a quadratic function. For instance, f (x) = x2 , where x ∈ Z, is an M -convex function with f • (p) = sup(px − x2 ) = sup(0, ±p − 1, ±2p − 4, ±3p − 9, . . .), x∈Z

which is not quadratic since f • (−1) = f • (0) = f • (1) = 0.

8.2

Duality

Discrete duality theorems lie at the heart of discrete convex analysis. Major theorems presented in this section are the separation theorem for M-convex/M-concave functions (M-separation theorem), the separation theorem for L-convex/L-concave functions (L-separation theorem), and the Fenchel-type duality theorem. These theorems look quite similar to the corresponding theorems in convex analysis, but they express, in fact, some deep facts of a combinatorial nature. Almost all duality results in optimization on matroids and submodular functions are corollaries of these theorems.

8.2.1

Separation Theorems

We start by reviewing the preliminary general discussion in section 1.2. A discrete separation theorem is a statement that, for f : ZV → Z ∪ {+∞} and h : ZV → Z ∪ {−∞} belonging to certain classes of functions, if f (x) ≥ h(x) for all x ∈ ZV , then there exist α∗ ∈ Z and p∗ ∈ ZV such that f (x) ≥ α∗ + p∗ , x ≥ h(x)

(∀ x ∈ ZV ).

(8.18)

Denoting by f the convex closure of f and by h the concave closure of h (i.e., −h is the convex closure of −h), we observed the following phenomena in Examples 1.5 and 1.6:  f (x) ≥ h(x) (∀ x ∈ RV ), 1. f (x) ≥ h(x) (∀ x ∈ ZV ) =⇒

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 existence of α∗ ∈ R and p∗ ∈ RV , 2. f (x) ≥ h(x) (∀ x ∈ ZV ) =⇒ ∗ ∗ 3. existence of α ∈ R and p ∈ RV =⇒  existence of α∗ ∈ Z and p∗ ∈ ZV . We will see below that all three implications hold true for M-convex and Lconvex functions. The following proposition addresses the first. Proposition 8.14. (1) If f, −h ∈ M [Z → R], then f (x) ≥ h(x) (∀ x ∈ ZV ) =⇒ f (x) ≥ h(x) (∀ x ∈ RV ). (2) If g, −k ∈ L [Z → R], then g(p) ≥ k(p) (∀ p ∈ ZV ) =⇒ g(p) ≥ k(p) (∀ p ∈ RV ). Proof. (1) Theorem 6.44 with f1 = f and f2 = −h shows that f1 + f2 ≥ 0 implies f1 + f2 ≥ 0. (2) It suffices to prove the claim when g, −k ∈ L[Z → R]. Theorem 7.19 applied to g and −k shows this. The separation theorem for M-convex/M-concave functions reads as follows. It should be clear that f • and h◦ are the convex and concave conjugate functions of f and h defined by (8.11) and (8.12), respectively. In the proof we use the notations  and ∂Z for the concave version of subdifferentials defined as ∂R  ∂R h(x) = −∂R (−h)(x),

∂Z h(x) = −∂Z (−h)(x).

(8.19)

Theorem 8.15 (M-separation theorem). Let f : ZV → R ∪ {+∞} be an M convex function and h : ZV → R ∪ {−∞} be an M -concave function such that domZ f ∩ domZ h = ∅ or domR f • ∩ domR h◦ = ∅. If f (x) ≥ h(x) (∀ x ∈ ZV ), there exist α∗ ∈ R and p∗ ∈ RV such that f (x) ≥ α∗ + p∗ , x ≥ h(x)

(∀ x ∈ ZV ).

(8.20)

Moreover, if f and h are integer valued, there exist integer-valued α∗ ∈ Z and p∗ ∈ ZV . Proof. We may assume f, −h ∈ M[Z → R]. (i) Suppose that domZ f ∩ domZ h = ∅. For the convex closure f of f and the concave closure h of h, we have f (x) ≥ h(x) (∀ x ∈ RV ) by Proposition 8.14 (1). Since domR f ∩ domR h = ∅, the separation theorem in convex analysis (Theorem 3.5) gives α∗ ∈ R and p∗ ∈ RV such that f (x) ≥ α∗ + p∗ , x ≥ h(x) (∀ x ∈ RV ) (see Note 8.19). This implies (8.20) since f = f and h = h on ZV by Theorem 6.42. The integrality assertion is proved from the facts that the integer subdifferential of an integer-valued M-convex function is an L-convex set and that Lconvex sets have the property of convexity in intersection. We may assume that inf{f (x) − h(x) | x ∈ ZV } = 0. Then there exists x0 ∈ ZV with f (x0 ) − h(x0 ) = 0

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(by the integrality of the function value). By (6.87) and Theorem 6.61 (2) we have   ∂R f (x0 ) ∩ ∂R h(x0 ) = ∂R f (x0 ) ∩ ∂R h(x0 ) = ∂Z f (x0 ) ∩ ∂Z h(x0 ),  which is nonempty since p∗ ∈ ∂R f (x0 ) ∩ ∂R h(x0 ). Since ∂Z f (x0 ) and ∂Z h(x0 ) above are L-convex, convexity in intersection for L-convex sets (5.9) guarantees the existence of an integer vector p∗∗ ∈ ∂Z f (x0 ) ∩ ∂Z h(x0 ). With this p∗∗ and α∗∗ = h(x0 ) − p∗∗ , x0 ∈ Z, the inequality (8.20) is satisfied. (ii) Next suppose that domZ f ∩ domZ h = ∅ and domR f • ∩ domR h◦ = ∅. For a fixed p0 ∈ domR f • ∩ domR h◦ and for any p ∈ RV , we have

f • (p) =

sup { p − p0 , x + [ p0 , x − f (x)]} ≤ x∈domZ f

h◦ (p) =

inf

x∈domZ h

{ p − p0 , x + [ p0 , x − h(x)]} ≥

from which follows + ◦



h (p)−f (p) ≥

sup p − p0 , x + f • (p0 ), x∈domZ f

p − p0 , x + h◦ (p0 ),

inf

x∈domZ h

, inf

x∈domZ h

p − p0 , x −

sup p − p0 , x +h◦ (p0 )−f • (p0 ). (8.21) x∈domZ f

Since domZ f and domZ h are disjoint M-convex sets, the separation theorem for Mconvex sets (Theorem 4.21) gives p∗ ∈ RV such that the right-hand side of (8.21) with p = p∗ is nonnegative. With this p∗ and α∗ ∈ R such that f • (p∗ ) ≤ −α∗ ≤ h◦ (p∗ ), the inequality (8.20) is satisfied. For integer-valued f and h, we have f • , −h◦ ∈ L[Z|R → R] and, hence, domR f • , domR h◦ ∈ L0 [Z|R]. We may assume p0 ∈ ZV by (5.9) and p∗ ∈ ZV by Theorem 4.21. Then f • (p∗ ) and h◦ (p∗ ) are integers, and therefore we can take an integer α∗ ∈ Z. Next we state the separation theorem for L-convex/L-concave functions. Theorem 8.16 (L-separation theorem). Let g : ZV → R ∪ {+∞} be an L convex function and k : ZV → R ∪ {−∞} be an L -concave function such that domZ g ∩ domZ k = ∅ or domR g • ∩ domR k ◦ = ∅. If g(p) ≥ k(p) (∀ p ∈ ZV ), there exist β ∗ ∈ R and x∗ ∈ RV such that g(p) ≥ β ∗ + p, x∗ ≥ k(p)

(∀ p ∈ ZV ).

(8.22)

Moreover, if g and k are integer valued, there exist integer-valued β ∗ ∈ Z and x∗ ∈ ZV . Proof. We may assume g, −k ∈ L[Z → R]. (i) Suppose that domZ g ∩ domZ k = ∅. For the convex closure g of g and the concave closure k of k, we have g(p) ≥ k(p) (∀ p ∈ RV ) by Proposition 8.14 (2). Since domR g ∩ domR k = ∅, the separation theorem in convex analysis (Theorem 3.5) gives β ∗ ∈ R and x∗ ∈ RV such that g(p) ≥ β ∗ + p, x∗ ≥ k(p) (∀ p ∈ RV ) (see Note 8.19). This implies (8.22) since g = g and k = k on ZV by Theorem 7.20.

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The integrality assertion is proved from the facts that the integer subdifferential of an integer-valued L-convex function is an M-convex set and that Mconvex sets have the property of convexity in intersection. We may assume that inf{g(p) − k(p) | p ∈ ZV } = 0. Then there exists p0 ∈ ZV with g(p0 ) − k(p0 ) = 0 (by the integrality of the function value). By (6.87) and Theorem 7.43 (2), we have   ∂R g(p0 ) ∩ ∂R k(p0 ) = ∂R g(p0 ) ∩ ∂R k(p0 ) = ∂Z g(p0 ) ∩ ∂Z k(p0 ),  k(p0 ). Since ∂Z g(p0 ) and ∂Z k(p0 ) which is nonempty since x∗ ∈ ∂R g(p0 ) ∩ ∂R above are M-convex, convexity in intersection for M-convex sets (4.34) guarantees the existence of an integer vector x∗∗ ∈ ∂Z g(p0 ) ∩ ∂Z k(p0 ). With this x∗∗ and β ∗∗ = k(p0 ) − p0 , x∗∗ ∈ Z, the inequality (8.22) is satisfied. (ii) Next suppose that domZ g ∩ domZ k = ∅ and domR g • ∩ domR k ◦ = ∅. For a fixed x0 ∈ domR g • ∩ domR k ◦ and for any x ∈ RV , we have

g • (x) =

sup { p, x − x0 + [ p, x0 − g(p)]} ≤ p∈domZ g

k ◦ (x) =

inf

p∈domZ k

{ p, x − x0 + [ p, x0 − k(p)]} ≥

from which follows + k ◦ (x)−g • (x) ≥

sup p, x − x0 + g • (x0 ), p∈domZ g

inf

p∈domZ k

p, x − x0 + k ◦ (x0 ),

, inf

p∈domZ k

p, x − x0 −

sup p, x − x0 +k ◦ (x0 )−g • (x0 ). (8.23) p∈domZ g

Since domZ g and domZ k are disjoint L-convex sets, the separation theorem for Lconvex sets (Theorem 5.9) gives x∗ ∈ RV such that the right-hand side of (8.23) with x = x∗ is nonnegative. With this x∗ and β ∗ ∈ R such that g • (x∗ ) ≤ −β ∗ ≤ k ◦ (x∗ ), the inequality (8.22) is satisfied. For integer-valued g and k we have g • , −k ◦ ∈ M[Z|R → R] and, hence, domR g • , domR k ◦ ∈ M0 [Z|R]. We may assume x0 ∈ ZV by (4.34) and x∗ ∈ ZV by Theorem 5.9. Then g • (x∗ ) and k ◦ (x∗ ) are integers, and therefore we can take an integer β ∗ ∈ Z. As an immediate corollary of the M-separation theorem we can obtain an optimality criterion for the problem of minimizing the sum of two M-convex functions, which we call the M-convex intersection problem. Note that the sum of M-convex functions is no longer M-convex and Theorem 6.26 (M-optimality criterion) does not apply. Theorem 8.17 (M-convex intersection theorem). For M -convex functions f1 , f2 ∈ M [Z → R] and a point x∗ ∈ domZ f1 ∩ domZ f2 , we have f1 (x∗ ) + f2 (x∗ ) ≤ f1 (x) + f2 (x)

(∀ x ∈ ZV )

(8.24)

if and only if there exists p∗ ∈ RV such that f1 [−p∗ ](x∗ ) ≤ f1 [−p∗ ](x) f2 [+p∗ ](x∗ ) ≤ f2 [+p∗ ](x)

(∀ x ∈ ZV ), (∀ x ∈ ZV ).

(8.25) (8.26)

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Conditions (8.25) and (8.26) are equivalent, respectively, to f1 [−p∗ ](x∗ ) ≤ f1 [−p∗ ](x∗ + χu − χv ) ∗







f2 [+p ](x ) ≤ f2 [+p ](x + χu − χv )

(∀ u, v ∈ V ∪ {0}),

(8.27)

(∀ u, v ∈ V ∪ {0}),

(8.28)

with the notation χ0 = 0, and for such a p∗ we have arg min(f1 + f2 ) = arg min f1 [−p∗ ] ∩ arg min f2 [+p∗ ].

(8.29)

Moreover, if f1 and f2 are integer valued, i.e., f1 , f2 ∈ M [Z → Z], we can choose integer-valued p∗ ∈ ZV . Proof. The sufficiency of (8.25) and (8.26) is obvious. Conversely, suppose that (8.24) is true and apply the M-separation theorem (Theorem 8.15) to f (x) = f1 (x) and h(x) = f1 (x∗ ) + f2 (x∗ ) − f2 (x) to obtain α∗ and p∗ satisfying (8.20). We have α∗ = f1 [−p∗ ](x∗ ) from (8.20) with x = x∗ and hence f1 (x) ≥ f1 [−p∗ ](x∗ ) + p∗ , x ≥ f1 (x∗ ) + f2 (x∗ ) − f2 (x). This implies (8.25) and (8.26), which are equivalent to (8.27) and (8.28), respectively, by the M-optimality criterion (Theorem 6.26 (2)). To prove (8.29) take x ˆ ∈ arg min(f1 + f2 ). Then f1 [−p∗ ](ˆ x) + f2 [+p∗ ](ˆ x) = f1 [−p∗ ](x∗ ) + f2 [+p∗ ](x∗ ), which, along with (8.25) and (8.26), implies x ˆ ∈ arg min f1 [−p∗ ] ∩ arg min f2 [+p∗ ]. Hence follows ⊆ in (8.29), whereas ⊇ is obvious. Finally, the integrality of p∗ is due to the integrality assertion in Theorem 8.15.

Note 8.18. The assumptions on the effective domains are necessary in the separation theorems (Theorems 8.15 and 8.16). For instance, for an M-convex function f : Z2 → Z ∪ {+∞} and an M-concave function h : Z2 → Z ∪ {−∞} defined by   x(1) (x(1) + x(2) = 1), −x(1) (x(1) + x(2) = −1), f (x) = h(x) = +∞ (otherwise), −∞ (otherwise), we have •

f (p) =



p(2) (p(1) − p(2) = 1), +∞ (otherwise),



h (p) =



−p(2) (p(1) − p(2) = −1), −∞ (otherwise),

domZ f ∩ domZ h = ∅, and domZ f • ∩ domZ h◦ = ∅. There exists no separating affine function for (f, h) or (f • , h◦ ). Note 8.19. This is a technical supplement to the proof of the M-separation theorem (Theorem 8.15). (A similar remark applies to the proof of the L-separation theorem.) We applied the separation theorem to f and h, the convex and concave

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extensions of f and h, without verifying the assumption in Theorem 3.5. If f and h are polyhedral, the assumption (a2) of Theorem 3.5 is met and the theorem is literally applicable. If inf{f (x) − h(x) | x ∈ ZV } is attained by some x = x0 ∈ ZV , we have 



f (x) ≥ f (x0 ) + f (x0 ; x − x0 ) ≥ h(x0 ) + h (x0 ; x − x0 ) ≥ h(x), 



in which the directional derivatives f (x0 ; ·) and h (x0 ; ·) of f and h at x0 are  polyhedral and Theorem 3.5 may be used for the pair of f (x0 ) + f (x0 ; x − x0 ) and  h(x0 ) + h (x0 ; x − x0 ). Otherwise we have to resort to a variant of the separation theorem such as the following: Let f : ZV → R ∪ {+∞} and h : ZV → R ∪ {−∞} be integrally convex and concave functions with domZ f ∩ domZ h = ∅, and denote their convex and concave extensions by f and h. If f (x) ≥ h(x) (∀ x ∈ RV ), then there exist α∗ ∈ R and p∗ ∈ RV such that f (x) ≥ α∗ + p∗ , x ≥ h(x) (∀ x ∈ RV ).

Note 8.20. The original proof of the M-separation theorem is based on an algorithmic argument for a generalization of the submodular flow problem involving an M-convex cost function (Murota [142]). In particular, the argument is purely discrete, not relying on the separation theorem in convex analysis. See sections 9.1.4 and 9.5.

8.2.2

Fenchel-Type Duality Theorem

The Fenchel-type duality theorem is discussed here. Before giving a precise statement of the theorem we explain the essence of the assertion. For any functions f : ZV → R ∪ {+∞} and h : ZV → R ∪ {−∞}, we have a chain of inequalities inf{f (x) − h(x) | x ∈ ZV } ≥ inf{f (x) − h(x) | x ∈ RV } ≥ sup{h◦ (p) − f • (p) | p ∈ RV } ≥ sup{h◦ (p) − f • (p) | p ∈ ZV }

(8.30)

from the definitions (8.11) and (8.12) of conjugate functions, where f and h are the convex and concave closures of f and h, respectively. We observe the following: 1. The second inequality is in fact an equality (under certain regularity assumptions) by the Fenchel duality theorem in convex analysis (Theorem 3.6). 2. The first inequality can be strict even when f is convex extensible and h is concave extensible, as is demonstrated by Example 1.6. A similar statement applies to the third inequality. The following theorem asserts that the first and third inequalities in (8.30) turn into equalities for M -convex/M-concave functions and L -convex/L-concave functions and that all three inequalities are equalities for such integer-valued functions.

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Theorem 8.21 (Fenchel-type duality theorem). (1) Let f : ZV → R∪{+∞} be an M -convex function and h : ZV → R∪{−∞} be an M -concave function, i.e., f, −h ∈ M [Z → R], such that domZ f ∩domZ h = ∅ or domR f • ∩ domR h◦ = ∅. Then we have inf{f (x) − h(x) | x ∈ ZV } = sup{h◦ (p) − f • (p) | p ∈ RV }.

(8.31)

If this common value is finite, the supremum is attained by some p ∈ domR f • ∩ domR h◦ . (2) Let g : ZV → R∪{+∞} be an L -convex function and k : ZV → R∪{−∞} be an L -concave function, i.e., g, −k ∈ L [Z → R], such that domZ g ∩ domZ k = ∅ or domR g • ∩ domR k ◦ = ∅. Then we have inf{g(p) − k(p) | p ∈ ZV } = sup{k ◦ (x) − g • (x) | x ∈ RV }.

(8.32)

If this common value is finite, the supremum is attained by some x ∈ domR g • ∩ domR k ◦ . (3) Let f : ZV → Z ∪ {+∞} be an integer-valued M -convex function and h : V Z → Z ∪ {−∞} be an integer-valued M -concave function, i.e., f, −h ∈ M [Z → Z], such that domZ f ∩ domZ h = ∅ or domZ f • ∩ domZ h◦ = ∅. Then we have inf{f (x) − h(x) | x ∈ ZV } = sup{h◦ (p) − f • (p) | p ∈ ZV }.

(8.33)

If this common value is finite, the infimum is attained by some x ∈ domZ f ∩ domZ h and the supremum is attained by some p ∈ domZ f • ∩ domZ h◦ . (4) Let g : ZV → Z ∪ {+∞} be an integer-valued L -convex function and k : V Z → Z ∪ {−∞} be an integer-valued L -concave function, i.e., g, −k ∈ L [Z → Z], such that domZ g ∩ domZ k = ∅ or domZ g • ∩ domZ k ◦ = ∅. Then we have inf{g(p) − k(p) | p ∈ ZV } = sup{k ◦ (x) − g • (x) | x ∈ ZV }.

(8.34)

If this common value is finite, the infimum is attained by some p ∈ domZ g ∩ domZ k and the supremum is attained by some x ∈ domZ g • ∩ domZ k ◦ . Proof. (1) Suppose that domZ f ∩ domZ h = ∅. By (8.30) we may assume that Δ = inf{f (x) − h(x) | x ∈ ZV } is finite. By the M-separation theorem (Theorem 8.15) for (f − Δ, h), there exist α∗ ∈ R and p∗ ∈ RV such that f (x) − Δ ≥ α∗ + p∗ , x ≥ h(x) for all x ∈ ZV , which implies h◦ (p∗ ) − f • (p∗ ) ≥ Δ. Combining this with (8.30) shows (8.31) as well as the attainment of the supremum by p∗ . Next suppose that domZ f ∩ domZ h = ∅ and domR f • ∩ domR h◦ = ∅. The separation theorem for M-convex sets (Theorem 4.21) applied to B1 = domZ h and B2 = domZ f gives p∗ ∈ {0, ±1}V satisfying (4.33). Putting p = p0 + cp∗ in (8.21) (within the proof of the M-separation theorem) and letting c → +∞, we obtain sup = +∞ in (8.31), whereas inf = +∞ by domZ f ∩ domZ h = ∅.

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(2) (The proof goes in parallel with (1).) Suppose that domZ g ∩ domZ k = ∅. By (8.30) we may assume that Δ = inf{g(p) − k(p) | p ∈ ZV } is finite. By the Lseparation theorem (Theorem 8.16) for (g − Δ, k), there exist β ∗ ∈ R and x∗ ∈ RV such that g(p) − Δ ≥ β ∗ + p, x∗ ≥ k(p) for all p ∈ ZV , which implies k ◦ (x∗ ) − g • (x∗ ) ≥ Δ. Combining this with (8.30) shows (8.32) as well as the attainment of the supremum by x∗ . Next suppose that domZ g ∩ domZ k = ∅ and domR g • ∩ domR k ◦ = ∅. The separation theorem for L-convex sets (Theorem 5.9) applied to D1 = domZ k and D2 = domZ g gives x∗ ∈ {0, ±1}V satisfying (5.10). Putting x = x0 + cx∗ in (8.23) (within the proof of the L-separation theorem) and letting c → +∞, we obtain sup = +∞ in (8.32), whereas inf = +∞ by domZ g ∩ domZ k = ∅. (3) In the proof of (1) we can take α∗ ∈ Z, p∗ ∈ ZV , and c ∈ Z. The supremum and infimum for finite (8.33) are attained since the functions are integer valued. (4) In the proof of (2) we can take β ∗ ∈ Z, x∗ ∈ ZV , and c ∈ Z. The supremum and infimum for finite (8.34) are attained since the functions are integer valued. The M-separation and L-separation theorems are parallel or conjugate in their statements as well as in their proofs. In contrast, the Fenchel-type duality theorem for integer-valued functions is self-conjugate in that the substitution of f = g • and h = k ◦ into (8.33) results in (8.34) by virtue of g = g •• and k = k ◦◦ . To emphasize the parallelism we have proved the M-separation theorem and the L-separation theorem independently and derived the Fenchel-type duality theorem therefrom. It is noted, however, that, with the knowledge of M-/L-conjugacy, these three duality theorems are almost equivalent to one another; once one of them is established, the other two can be derived by relatively easy formal calculations. Note 8.22. In Theorem 8.21 (1) the infimum is not necessarily attained by any x ∈ ZV (and similarly for (2)). For example, consider f : Z → R ∪ {+∞} and h : Z → R ∪ {−∞} defined by  f (x) =

exp(−x) +∞

(x ≥ 0), (x < 0),

 h(x) =

0 (x ≥ 0), −∞ (x < 0),

which are M -convex and M -concave, respectively. We have domZ f = domZ h = Z+ , domR f • = (−∞, 0]R , domR h◦ = [0, +∞)R , and inf = sup = 0 in (8.31). However, no x attains the infimum, whereas the supremum is attained by p = 0. Note 8.23. The assumptions on the effective domains are necessary in Theorem 8.21. For the M-convex and M-concave functions f and h in Note 8.18, we have domZ f ∩ domZ h = ∅ and domZ f • ∩ domZ h◦ = ∅. The identity (8.33) fails with infimum = +∞ and supremum = −∞.

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M-separation theorem f (x) ≥ α∗ + p∗ , x ≥ h(x) Fenchel-type duality (Fujishige [62]) ! Intersection theorem (Edmonds [44]) ⎧ ! ⎪ ⎪ ⎪ ⇒ Discrete separation for submodular functions Fenchel-type duality ⎪ ⎨ (Frank [55]) inf{f − h} ⎪ ⎪ ⇒ Valuated matroid intersection = sup{h◦ − f • } ⎪ ⎪ ⎩ (Murota [135]) ! ⇓ L-separation theorem Weighted matroid intersection f • (p) ≥ β ∗ + p, x∗ ≥ h◦ (p) (Edmonds [45], Frank [54], Iri–Tomizawa [96]) Figure 8.2. Duality theorems (f : M -convex function, h: M -concave function).

8.2.3

Implications

In spite of the apparent similarity to the corresponding theorems in convex analysis, the discrete duality theorems established above convey deep combinatorial properties of M-convex and L-convex functions. We now demonstrate this by deriving major duality results in optimization on matroids and submodular functions as immediate corollaries of these theorems (see also Fig. 8.2). The connection to the duality in network flow problems is discussed in Chapter 9. Example 8.24. Frank’s discrete separation theorem (Theorem 4.17) is a special case of the L-separation theorem (Theorem 8.16). By Proposition 7.4, the submodular and supermodular set functions ρ and μ can be identified, respectively, with an L -convex function g : ZV → R ∪ {+∞} with domZ g ⊆ {0, 1}V and an L -concave function k : ZV → R ∪ {−∞} with domZ k ⊆ {0, 1}V by ρ(X) = g(χX ) and μ(X) = k(χX ) for X ⊆ V . The L-separation theorem applies to (g, k) since the first assumption, domZ g ∩ domZ k = ∅, is met by g(0) = k(0) = 0, which follows from ρ(∅) = μ(∅) = 0. We see β ∗ = 0 from the inequality (8.22) for p = 0, and then the desired inequality (4.27) is obtained from (8.22) with p = χX for X ⊆ V . When ρ and μ are integer valued, g and k are also integer valued, and the integrality assertion in the L-separation theorem implies the integrality assertion in Theorem 4.17. Example 8.25. Edmonds’s intersection theorem (Theorem 4.18) in the integral case is a special case of the Fenchel-type duality theorem (Theorem 8.21 (3)). This is explained in Example 1.20. Example 8.26. The Fenchel-type duality theorem for submodular set functions is a special case of the Fenchel-type duality theorem for L -convex functions (Theorem 8.21 (2), (4)). The conjugate functions of a submodular set function ρ : 2V → R∪{+∞} and a supermodular set function μ : 2V → R∪{−∞} (i.e., ρ, −μ ∈ S[R])

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are defined by ρ• (x) = max{x(X) − ρ(X) | X ⊆ V } μ◦ (x) = min{x(X) − μ(X) | X ⊆ V }

(x ∈ RV ), (x ∈ RV ).

The Fenchel-type duality theorem for submodular set functions is an identity min{ρ(X) − μ(X) | X ⊆ V } = max{μ◦ (x) − ρ• (x) | x ∈ RV }

(8.35)

with an additional integrality assertion that, for integer-valued ρ and μ, the maximum on the right-hand side of (8.35) can be attained by an integer vector x ∈ ZV . As in Example 8.24, we consider an L -convex function g and an L -concave function k associated with ρ and μ. We have g • = ρ• , k ◦ = μ◦ , and domZ g ∩ domZ k = ∅, and, therefore, (8.35) is obtained as a special case of (8.32) and (8.34). Example 8.27. Frank’s weight-splitting theorem for the weighted matroid intersection problem is a special case of the optimality criterion for the M-convex intersection problem (Theorem 8.17). Given two matroids (V, B1 ) and (V, B2 ) on a common ground set V with base families B1 and B2 , as well as a weight vector w : V → R, the optimal common base problem is to find B ∈ B1 ∩ B2 that minimizes the weight w(B) = v∈B w(v). Frank’s weight-splitting theorem says that a common base B ∗ ∈ B1 ∩ B2 is optimal if and only if there exist real vectors w1∗ , w2∗ : V → R such that (i) w = w1∗ + w2∗ , (ii) B ∗ is a minimum-weight base of (V, B1 ) with respect to w1∗ , and (iii) B ∗ is a minimum-weight base of (V, B2 ) with respect to w2∗ . In addition, the theorem states that, if w is integer valued, the vectors w1∗ and w2∗ can be chosen to be integer valued. The combinatorial content of this theorem lies in the assertion about the existence of an integer weight splitting in the case of integer-valued weight. Applying Theorem 8.17 to a pair of M-convex functions   w(B) (x = χB , B ∈ B1 ), 0 (x = χB , B ∈ B2 ), f2 (x) = f1 (x) = +∞ (otherwise), +∞ (otherwise) yields p∗ satisfying (8.25) and (8.26) with additional integrality in the case of integervalued w. A weight splitting constructed by w1∗ = w − p∗ ,

w2∗ = p∗

has the properties (ii) and (iii) because of (8.25) and (8.26). In Example 1.21 we derived the weight-splitting theorem (integer-weight case) from the M-separation theorem (Theorem 8.15). Example 8.28. Suppose we are given two valuated matroids (V, ω1 ) and (V, ω2 ) as well as a weight vector w : V → R. The valuated matroid intersection problem is to find B ⊆ V that maximizes w(B) + ω1 (B) + ω2 (B). The weight-splitting theorem for valuated matroid intersection says that a common base B ∗ maximizes

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w(B) + ω1 (B) + ω2 (B) if and only if there exist real vectors w1∗ , w2∗ : V → R such that (i) w = w1∗ + w2∗ , (ii) B ∗ maximizes ω1 [w1∗ ], and (iii) B ∗ maximizes ω2 [w2∗ ], where ω1 [w1∗ ] and ω2 [w2∗ ] are defined by (2.76). In addition, the theorem states that, if ω1 , ω2 , and w are all integer valued, the vectors w1∗ and w2∗ can be chosen to be integer valued. Let f1 and f2 be the M-convex functions associated, respectively, with ω1 and ω2 by (2.77). Maximizing w(B) + ω1 (B) + ω2 (B) is equivalent to minimizing f1 (x) + f2 [−w](x), and a desired weight splitting can be obtained from the M-convex intersection theorem (Theorem 8.17) as in Example 8.27. It is emphasized again that the discrete duality theorems are of combinatorial nature and cannot be obtained through mere combination of the convex-extensibility theorem (Theorems 6.42 and 7.20) with the separation theorem (Theorem 3.5) or the Fenchel duality theorem (Theorem 3.6) for (ordinary) convex functions. Examples 1.5 and 1.6 should be convincing enough to demonstrate this point.

8.3

M2 -Convex Functions and L2 -Convex Functions

Two additional classes of discrete functions, called M2 -convex functions and L2 convex functions, are considered here. An M2 -convex function is a function representable as the sum of two M-convex functions, and an L2 -convex function is the integer infimal convolution of two L-convex functions. These functions play crucial roles in combinatorial optimization. In Edmonds’s intersection theorem (Theorem 4.18), for example, the left-hand side of the min-max relation (4.29) corresponds to M2 -convexity and the right-hand side to L2 -convexity.

8.3.1

M2 -Convex Functions

A function f : ZV → R ∪ {+∞} with dom f = ∅ is said to be M2 -convex if it can be represented as the sum of two M-convex functions, i.e., if f = f1 + f2 for some f1 , f2 ∈ M[Z → R]. We denote by M2 [Z → R] the set of M2 -convex functions and by M2 [Z → Z] the subclass of M2 -convex functions f = f1 + f2 with some f1 , f2 ∈ M[Z → Z]. An M2 -convex function is defined similarly as the sum of two M -convex functions, which is obtained as the projection of an M2 -convex function. The notations M2 [Z → R] and M2 [Z → Z] are defined in an obvious way. We have



M [Z → R]

M[Z → Z]

⊂ M2 [Z → Z] ⊂

M2 [Z → Z] ⊂

M[Z → R]

M2 [Z → R] ⊂



M2 [Z → R] ⊂



(8.36)

M [Z → Z]

Note that a set is M2 -convex (resp., M2 -convex) if and only if its indicator function is M2 -convex (resp., M2 -convex). The effective domain and the set of minimizers of an M2 -convex function are M2 -convex sets; the latter is a consequence of the M-convex intersection theorem (Theorem 8.17).

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Proposition 8.29. (1) For an M2 -convex function f , dom f is M2 -convex. (2) For an M2 -convex function f , dom f is M2 -convex. Proof. This follows from the relation dom (f1 + f2 ) = dom f1 ∩ dom f2 and the Mor M -convexity of dom fi (i = 1, 2) given in Proposition 6.7.

Proposition 8.30. (1) For an M2 -convex function f , arg min f is M2 -convex if it is not empty. (2) For an M2 -convex function f , arg min f is M2 -convex if it is not empty. Proof. This follows from arg min(f1 + f2 ) = arg min f1 [−p∗ ] ∩ arg min f2 [+p∗ ] in (8.29) and the M- or M -convexity of arg min f1 [−p∗ ] and arg min f2 [+p∗ ] given in Proposition 6.29. M2 -convexity implies integral convexity. Theorem 8.31. An M2 -convex function is integrally convex. In particular, an M2 -convex set is integrally convex. Proof. For f = f1 + f2 with f1 , f2 ∈ M [Z → R] and x ∈ RV , Theorem 6.44 implies f˜(x) = f˜1 (x) + f˜2 (x) = f1 (x) + f2 (x), where f˜, f˜1 , and f˜2 are the local convex extensions (3.61) of f , f1 , and f2 , respectively, and f1 and f2 are the convex closures (3.56) of f1 and f2 . Since f1 + f2 is convex, so is f˜. For the minimality of an M2 -convex function we have the following criterion. Theorem 8.32 (M2 -optimality criterion). For an M2 -convex function f ∈ M2 [Z → R] and x ∈ dom f , we have f (x) ≤ f (y) (∀ y ∈ ZV ) ⇐⇒ f (x) ≤ f (x + χY − χZ ) (∀ Y, Z ⊆ V, |Y | = |Z|). Proof. By Theorem 8.31 the optimality criterion for an integrally convex function (Theorem 3.21) applies. We may impose the condition |Y | = |Z| because x(V ) is constant for any x ∈ dom f . The optimality criterion above is not suitable for polynomial-time verification. If the summands f1 and f2 in f = f1 + f2 are known, the minimality can be verified in polynomial time by the following criterion, as will be explained in Note 9.21. We mention that the M-convex intersection theorem (Theorem 8.17) also serves as an optimality criterion for M2 -convex functions when the summands are known.

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Theorem 8.33 (M2 -optimality criterion). For M-convex functions f1 , f2 ∈ M[Z → R] and a point x ∈ dom f1 ∩ dom f2 , we have f1 (x) + f2 (x) ≤ f1 (y) + f2 (y)

(∀ y ∈ ZV )

if and only if k 

[f1 (x − χui + χvi ) − f1 (x)] +

i=1

k 

[f2 (x + χui+1 − χvi ) − f2 (x)] ≥ 0

i=1

for any u1 , . . . , uk , v1 , . . . , vk ∈ V with {u1 , . . . , uk }∩{v1 , . . . , vk } = ∅, where uk+1 = u1 by convention. Proof. The proof is given later in Note 9.21. A scaling version of the optimality criterion above leads to a proximity theorem for M2 -convex functions. Theorem 8.34 (M2 -proximity theorem). Let f1 , f2 ∈ M[Z → R] be M-convex functions, and assume α ∈ Z++ and n = |V |. If xα ∈ dom f1 ∩ dom f2 satisfies k 

[f1 (xα − α(χui − χvi )) − f1 (xα )] +

i=1

k 

[f2 (xα + α(χui+1 − χvi )) − f2 (xα )] ≥ 0

i=1

for any u1 , . . . , uk , v1 , . . . , vk ∈ V with {u1 , . . . , uk }∩{v1 , . . . , vk } = ∅, where uk+1 = u1 , then arg min(f1 + f2 ) = ∅ and there exists x∗ ∈ arg min(f1 + f2 ) with ||xα − x∗ ||∞ ≤

n2 (α − 1). 2

(8.37)

Proof. See Murota–Tamura [162]. Straightforward calculations based on the M-convex intersection theorem yield the following two theorems. Theorem 8.35. (1) For f1 , f2 ∈ M [Z → R] and x ∈ dom f1 ∩ dom f2 , we have ∂R (f1 + f2 )(x) = ∂R f1 (x) + ∂R f2 (x) = ∅. (2) For f1 , f2 ∈ M [Z → Z] and x ∈ dom f1 ∩ dom f2 , we have ∂Z (f1 + f2 )(x) = ∂Z f1 (x) + ∂Z f2 (x) = ∅. (3) For f ∈ M2 [Z → Z] and x ∈ dom f , ∂Z f (x) is an L2 -convex set. For f ∈ M2 [Z → Z] and x ∈ dom f , ∂Z f (x) is an L2 -convex set.

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Proof. Using the M-convex intersection theorem (Theorem 8.17), we see that p ∈ ∂R (f1 + f2 )(x) ⇐⇒ x ∈ arg min(f1 + f2 [−p]) ⇐⇒ ∃ q ∈ RV : x ∈ arg min f1 [−q] ∩ arg min f2 [−p + q] ⇐⇒ ∃ q ∈ RV : q ∈ ∂R f1 (x) and p − q ∈ ∂R f2 (x) ⇐⇒ p ∈ ∂R f1 (x) + ∂R f2 (x). For (2) and (3), note that ∂R fi (x) ∈ L0 [Z|R] from Theorem 6.61 (2). For any f1 , f2 : ZV → Z ∪ {+∞}, (f1 2Z f2 )• = f1 • + f2 • ,

(8.38)

where 2Z denotes the integer infimal convolution (6.43) and • is the discrete Legendre–Fenchel transformation (8.11)Z . A relation conjugate to this also holds for M-convex functions as follows. Theorem 8.36. For integer-valued M -convex functions f1 , f2 ∈ M [Z → Z] with dom f1 ∩ dom f2 = ∅, we have (f1 + f2 )• = f1 • 2Z f2 • and (f1 + f2 )•• = f1 + f2 . Proof. For p ∈ dom (f1 + f2 )• there exists q ∈ ZV such that (f1 +f2 )• (p) = − min(f1 +f2 [−p]) = − min f1 [−q]−min f2 [−p+q] = f1• (q)+f2• (p−q) by the M-convex intersection theorem (Theorem 8.17). This shows that (f1 +f2 )• ≥ f1 • 2Z f2 • , whereas ≤ is obvious from p, x − f1 (x) − f2 (x) ≤ f1• (q) + f2• (p − q). The second identity follows from the first because of (8.38) and Theorem 8.12.

8.3.2

L2 -Convex Functions

A function g : ZV → R ∪ {+∞} is said to be L2 -convex if it can be represented as the infimal convolution of two L-convex functions, i.e., if g = g1 2Z g2 for some g1 , g2 ∈ L[Z → R]. We denote by L2 [Z → R] the set of L2 -convex functions and by L2 [Z → Z] the subclass of L2 -convex functions g = g1 2Z g2 with some g1 , g2 ∈ L[Z → Z]. An L2 -convex function is defined similarly as the integer infimal convolution of two L -convex functions, which is obtained as the restriction of an L2 -convex function. The notations L2 [Z → R] and L2 [Z → Z] are defined in an obvious way. We have



L [Z → R]

L[Z → Z]

⊂ L2 [Z → Z] ⊂

L2 [Z → Z] ⊂

L[Z → R]

L2 [Z → R] ⊂



L2 [Z → R] ⊂



(8.39)

L [Z → Z]

Note that a set is L2 -convex (resp., L2 -convex) if and only if its indicator function is L2 -convex (resp., L2 -convex).

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Note 8.37. Here is a technical supplement concerning the definition of an L2 convex function. By definition, a function g : ZV → R ∪ {+∞} is L2 -convex if it can be represented as g(p) = inf{ˆ g1 (p1 ) + gˆ2 (p2 ) | p = p1 + p2 ; p1 , p2 ∈ ZV }

(8.40)

for some gˆ1 , gˆ2 ∈ L[Z → R]. We may assume that the infimum is attained for each p ∈ dom g. Namely, it is known that for an L2 -convex function g there exist L-convex functions g1 , g2 ∈ L[Z → R] such that g(p) = min{g1 (p1 ) + g2 (p2 ) | p = p1 + p2 ; p1 , p2 ∈ ZV }

(p ∈ dom g).

(8.41)

As an example, consider a pair of L-convex functions in two variables: gˆ1 (p) = exp(p(2) − p(1)),

gˆ2 (p) = exp(p(1) − p(2)).

The infimal convolution g = gˆ1 2Z gˆ2 is identically zero, with the infimum in (8.40) unattained. An obvious valid choice for (8.41) is g1 = g2 = 0 (identically). Note 8.38. For g1 , g2 ∈ L[Z → R], it can be shown that (g1 2Z g2 )(p0 ) = −∞ (∃ p0 ) =⇒ (g1 2Z g2 )(p) = −∞ (∀ p ∈ dom g1 + dom g2 ). The effective domain and the set of minimizers of an L2 -convex function are L2 -convex sets. Proposition 8.39. (1) For an L2 -convex function g, dom g is L2 -convex. (2) For an L2 -convex function g, dom g is L2 -convex. Proof. This follows from the relation dom (g1 2Z g2 ) = dom g1 + dom g2 and the Lor L -convexity of dom gi (i = 1, 2) given in Proposition 7.8.

Proposition 8.40. (1) For an L2 -convex function g, arg min g is L2 -convex if it is not empty. (2) For an L2 -convex function g, arg min g is L2 -convex if it is not empty. Proof. By (8.41) this follows from a general fact in Proposition 8.41 below and the L- or L -convexity of arg min gi (i = 1, 2) given in Proposition 7.16. Proposition 8.41. If g1 , g2 : ZV → R ∪ {+∞} are such that for every p ∈ ZV , inf{g1 (p1 ) + g2 (p2 ) | p = p1 + p2 ; p1 , p2 ∈ ZV } is attained whenever it is finite,

(8.42)

then we have arg min(g1 2Z g2 )[−x] = arg min g1 [−x] + arg min g2 [−x]

(∀ x ∈ RV ).

(8.43)

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Proof. It suffices to prove arg min(g1 2Z g2 )[−x] ⊆ arg min g1 [−x] + arg min g2 [−x], since the converse inclusion is always true, independently of (8.42). Take p∗ ∈ arg min(g1 2Z g2 )[−x]. By (8.42) there exist p∗1 and p∗2 such that p∗ = p∗1 + p∗2 and (g1 2Z g2 )(p∗ ) = g1 (p∗1 ) + g2 (p∗2 ). If g1 [−x](p∗1 ) > g1 [−x](p1 ) for some p1 , we would have (g1 2Z g2 )[−x](p∗ ) = g1 [−x](p∗1 ) + g2 [−x](p∗2 ) > g1 [−x](p1 ) + g2 [−x](p∗2 ) ≥ (g1 2Z g2 )[−x](p1 + p∗2 ), a contradiction to the choice of p∗ . Hence p∗1 ∈ arg min g1 [−x]. Similarly, we have p∗2 ∈ arg min g2 [−x]. The assumption (8.42) above is necessary for the identity (8.43) to hold. For g1 2Z gˆ2 ) = Z2 the functions gˆ1 and gˆ2 in Note 8.37, for instance, we have arg min(ˆ but arg min gˆ1 = arg min gˆ2 = ∅. For integer-valued functions, however, (8.42) is always satisfied. L2 -convexity implies integral convexity. Theorem 8.42. An L2 -convex function is integrally convex. In particular, an L2 -convex set is integrally convex. Proof. It suffices to consider L2 -convex sets and functions. To emphasize the essence we give a proof for the integral convexity of an L2 -convex set. This implies, by Theorem 3.29, the integral convexity of an L2 -convex function g with a bounded effective domain, since arg min g[−x] is an L2 -convex set for any x ∈ RV by Proposition 8.40. A complete proof can be found in Murota–Shioura [153]. Let S be an L2 -convex set represented as S = D1 + D2 with D1 , D2 ∈ L0 [Z]. We will show that p ∈ S ⇒ p ∈ S ∩ N (p) (see (3.71)). By S = D1 + D2 = D1 + D2 we have p = p1 + p2 for some p1 ∈ D1 and p2 ∈ D2 . Put a1 = p1 − p1  and a2 = p2 − p2 , where 0 ≤ ak (v) < 1 for k = 1, 2 and v ∈ V . Denoting the distinct values among {a1 (v), a2 (v) | v ∈ V } by α1 > α2 > · · · > αm (≥ 0) and defining Uki = {v ∈ V | ak (v) ≥ αi } for k = 1, 2 and i = 1, . . . , m, we have ak =

m−1 

(αi − αi+1 )χUki + αm χUkm

(k = 1, 2),

i=1

and, hence, p = p1 + p2 =

m 

(αi − αi+1 )( p1  + χU1i + p2 − χU2i ),

i=0

where α0 = 1, αm+1 = 0, and U10 = U20 = ∅. This implies p ∈ S ∩ N (p), since qi = p1  + χU1i + p2 − χU2i belongs to S ∩ N (p) for i = 0, 1, . . . , m, as shown below.

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[Proof of qi ∈ S] We have p1  + χU1i ∈ D1 by Theorem 5.10 for p1 ∈ D1 . Since −p2 ∈ −D2 , we similarly see p2 − χU2i ∈ D2 . Hence qi ∈ D1 + D2 = S. [Proof of qi ∈ N (p)] We are to show p(v) ∈ Z =⇒ qi (v) = p(v),

(8.44)

p(v) ∈ / Z =⇒ p(v) ≤ qi (v) ≤ p(v) + 1.

(8.45)

By p1  + p2 = p − a1 + a2 ∈ ZV and −1 < −a1 (v) + a2 (v) < 1 (v ∈ V ), we have p(v) ∈ Z =⇒ p1 (v) + p2 (v) = p(v), a1 (v) = a2 (v), p(v) ∈ / Z =⇒ p1 (v) + p2 (v) ∈ { p(v), p(v) + 1}.

(8.46) (8.47)

/ Z. If p(v) ∈ Z, (8.46) shows χU1i (v) = χU2i (v), which implies (8.44). Suppose p(v) ∈ We put W = {v ∈ V | p1 (v) + p2 (v) = p(v) + 1} and divide into two cases: (i) v ∈ W and (ii) v ∈ V \ W . In case (i), let i be such that v ∈ U1i . Then v ∈ U2i follows from a2 (v) = p1 (v) + p2 (v) − p(v) + a1 (v) = p(v) + 1 − p(v) + a1 (v) ≥ a1 (v) ≥ αi . Therefore, −1 ≤ χU1i (v) − χU2i (v) ≤ 0, which implies (8.45). In case (ii), let i be such that v ∈ U2i . Then v ∈ U1i follows from a1 (v) = − p1 (v) − p2 (v) + p(v) + a2 (v) = − p(v) + p(v) + a2 (v) ≥ a2 (v) ≥ αi . Therefore, 0 ≤ χU1i (v) − χU2i (v) ≤ 1, which implies (8.45). For the minimality of an L2 -convex function, we have the following criterion. Theorem 8.43 (L2 -optimality criterion). For an L2 -convex function g ∈ L2 [Z → R] and p ∈ dom g, we have  g(p) ≤ g(p + χY ) (∀ Y ⊆ V ), g(p) ≤ g(q) (∀ q ∈ ZV ) ⇐⇒ (8.48) g(p) = g(p + 1). Proof. By Theorem 8.42 this is obtained as a special case of the optimality criterion for an integrally convex function (Proposition 3.22). A scaling version of the optimality criterion above leads to a proximity theorem for L2 -convex functions. Theorem 8.44 (L2 -proximity theorem). Let g : ZV → R ∪ {+∞} be an L2 -convex function such that g(p) = g(p + 1) (∀ p ∈ ZV ), and assume α ∈ Z++ and n = |V |. If pα ∈ dom g satisfies g(pα ) ≤ g(pα + αχY )

(∀ Y ⊆ V ),

(8.49)

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233

then arg min g = ∅ and there exists p∗ ∈ arg min g with pα ≤ p∗ ≤ pα + 2(n − 1)(α − 1)1.

(8.50)

Proof. Let g be represented as (8.41) with L-convex functions g1 and g2 , where gi (p + 1) = gi (p) (∀ p) for i = 1, 2 as a consequence of g(p) = g(p + 1) (∀ p). There α V α α α α exist pα such that g(pα ) = g1 (pα 1 , p2 ∈ Z 1 ) + g2 (p2 ) and p = p1 + p2 . For any Y ⊆ V we have α α α g(pα + αχY ) ≤ min {g1 (pα 1 + αχY ) + g2 (p2 ), g1 (p1 ) + g2 (p2 + αχY )} α α α by the definition of infimal convolution, whereas g1 (pα 1 ) + g2 (p2 ) = g(p ) ≤ g(p + αχY ) by (8.49). Hence α g1 (pα 1 ) ≤ g1 (p1 + αχY ),

α g2 (pα 2 ) ≤ g2 (p2 + αχY )

(∀ Y ⊆ V ).

By the L-proximity theorem (Theorem 7.18) there exist p∗1 ∈ arg min g1 and p∗2 ∈ arg min g2 such that ∗ α pα 1 ≤ p1 ≤ p1 + (n − 1)(α − 1)1,

∗ α pα 2 ≤ p2 ≤ p2 + (n − 1)(α − 1)1.

Then p∗ = p∗1 + p∗2 satisfies (8.50). Moreover, p∗ is a minimizer of g because p∗1 ∈ arg min g1 and p∗2 ∈ arg min g2 . The following two theorems are the counterparts of Theorems 8.35 and 8.36. Theorem 8.45. (1) For g1 , g2 ∈ L [Z → R] with g1 2Z g2 > −∞ and (8.42) and p ∈ dom (g1 2Z g2 ), there exist pi ∈ dom gi (i = 1, 2) such that p = p1 + p2 and ∂R (g1 2Z g2 )(p) = ∂R g1 (p1 ) ∩ ∂R g2 (p2 ) = ∅. (2) For g1 , g2 ∈ L [Z → Z] with g1 2Z g2 > −∞ and p ∈ dom (g1 2Z g2 ), there exist pi ∈ dom gi (i = 1, 2) such that p = p1 + p2 and ∂Z (g1 2Z g2 )(p) = ∂Z g1 (p1 ) ∩ ∂Z g2 (p2 ) = ∅. (3) For g ∈ L2 [Z → Z] and p ∈ dom g, ∂Z g(p) is an M2 -convex set. For g ∈ L2 [Z → Z] and p ∈ dom g, ∂Z g(p) is an M2 -convex set. Proof. Recall the relation x ∈ ∂R (g1 2Z g2 )(p) ⇔ p ∈ arg min(g1 2Z g2 )[−x] from (3.30), and use (8.43) and ∂R gi (pi ) ∈ M0 [Z|R], which is a variant of Theorem 7.43 (2). Theorem 8.46. For integer-valued L -convex functions g1 , g2 ∈ L [Z → Z] with g1 2Z g2 > −∞, we have (g1 2Z g2 )•• = g1 2Z g2 , where • means the discrete Legendre–Fenchel transformation (8.11)Z .

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Proof. Applying Theorem 8.36 to fi = gi • ∈ M [Z → Z] shows this. Note 8.47. An L2 -convex function g(p) = min{g1 (q) + g2 (p − q) | q ∈ ZV }, represented as in (8.41), can be evaluated efficiently, since g1 (q) + g2 (p − q) is an L-convex function in q, to which the minimization algorithms in section 10.3 can be applied.

8.3.3

Relationship

The relationship between M2 - and L2 -convex functions is discussed here. The first theorem shows the conjugacy relationship between M2 - and L2 -convex functions. Theorem 8.48. The two classes of functions M2 [Z → Z] and L2 [Z → Z] are in one-to-one correspondence under the discrete Legendre–Fenchel transformation (8.11)Z , and similarly for M2 [Z → Z] and L2 [Z → Z]. Proof. This is due to (8.38) and Theorems 8.36, 8.46, and 8.12. Separable convex functions are characterized as functions possessing both M2 convexity and L2 -convexity. Theorem 8.49. For a function f : ZV → R ∪ {+∞}, we have f is M2 -convex and L2 -convex ⇐⇒ f is M -convex and L -convex ⇐⇒ f is separable convex. Proof. It suffices to show that, if f is both M2 -convex and L2 -convex, then it is separable convex. We may assume that dom f is bounded. Take any p ∈ RV . By Propositions 8.30 and 8.40, the set arg min f [−p] is both M2 -convex and L2 -convex, and therefore it is an integer interval. This means that f is a separable convex function.

8.4 8.4.1

Lagrange Duality for Optimization Outline

On the basis of the conjugacy and duality theorems we can develop a Lagrange duality theory for a (nonlinear) integer program: P:

Minimize c(x)

subject to

x ∈ B,

(8.51)

where c : ZV → Z ∪ {+∞} and ∅ = B ⊆ ZV . The canonical “convex” case consists of problems in which

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(REG) B is an M-convex set, and (OBJ) c is an M-convex function. We refer to a problem with (REG) and (OBJ) as an M-convex program. We follow Rockafellar’s conjugate duality approach [177] to convex/nonconvex programs in nonlinear optimization. The whole scenario of the present section is a straightforward adaptation of it, whereas the technical development leading to a strong duality assertion for “convex” programs relies heavily on fundamental theorems of a combinatorial nature. An adaptation of the Lagrangian function in nonlinear programming affords a duality framework that covers “nonconvex” programs. We follow the notation of [177] to emphasize the parallelism. In the canonical “convex” case, the problem dual to P turns out to be a maximization of an L2 -concave function, where the strong duality holds between the pair of primal/dual problems. This is a consequence of the conjugacy between M2 and L2 -convexity and the Fenchel-type duality theorem for M-/L-convex functions. In the literature of integer programming we can find a number of duality frameworks, such as the subadditive duality. The present approach is distinguished from those in the following ways: 1. It is primarily concerned with nonlinear objective functions. 2. The theory parallels the perturbation-based duality formalism in nonlinear programming. 3. In particular, the dual problem is derived from an embedding of the given problem in a family of perturbed problems with a certain convexity in the direction of perturbation. 4. It identifies M-convex programs as the well-behaved core structure to be compared to convex programs in nonlinear programming.

8.4.2

General Duality Framework

We describe the general framework, in which neither (REG) nor (OBJ) is assumed. First we rewrite the problem P as follows: P:

Minimize f (x)

subject to x ∈ ZV

(8.52)

with f (x) = c(x) + δB (x),

(8.53)

where δB : ZV → {0, +∞} is the indicator function of B. We say that the problem P is feasible if f (x) < +∞ for some x ∈ ZV . Next we embed the optimization problem P in a family of perturbed problems. As the perturbation of f we consider F : ZV × ZU → Z ∪ {+∞}, with U being a finite set, such that F (x, 0) = f (x), F (x, ·)•• = F (x, ·) for each x ∈ ZV .

(8.54) (8.55)

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Here the second condition (8.55) means that the integer biconjugate of F (x, u) as a function in u for each fixed x coincides with F (x, u) itself. Note 8.50. By Proposition 8.11, the condition (8.55) is satisfied if, for each x, either F (x, ·) ≡ +∞ or the integer subdifferential of F (x, u) with respect to u is nonempty for each u ∈ dom F (x, ·). Recall that an integer-valued M2 - or L2 -convex function has a nonempty integer subdifferential (Theorems 8.35 and 8.45). The resulting family of optimization problems, parametrized by u ∈ ZU , reads as follows: P(u):

Minimize F (x, u)

subject to x ∈ ZV .

(8.56)

We define the optimal value function ϕ : ZU → Z ∪ {±∞} by ϕ(u) = inf{F (x, u) | x ∈ ZV }

(u ∈ ZU )

(8.57)

and the Lagrangian function K : ZV × ZU → Z ∪ {±∞} by K(x, y) = inf{F (x, u) + u, y | u ∈ ZU }

(x ∈ ZV , y ∈ ZU ).

(8.58)

For each x ∈ ZV , the function K(x, ·) : y → K(x, y) is the concave discrete Legendre–Fenchel transform of the function −F (x, ·) : u → −F (x, u). Our assumptions (8.54) and (8.55) on F (x, u) guarantee the following. Proposition 8.51. (1) F (x, u) = sup{K(x, y) − u, y | y ∈ ZU } (2) f (x) = sup{K(x, y) | y ∈ ZU } (x ∈ ZV ).

(x ∈ ZV , u ∈ ZU ).

Proof. (1) Abbreviate F (x, u) and K(x, y) to F (u) and K(y), respectively. We have F • (y) = −K(−y) by (8.58), while F (u) = F •• (u) by (8.55). Therefore, F (u) = F •• (u) = sup( u, y + K(−y)) = sup(K(y) − u, y ). y

y

(2) This follows from (1) with u = 0 and (8.54). We define the dual problem to P as follows: D:

Maximize g(y) subject to y ∈ ZU ,

(8.59)

where the objective function g : ZU → Z ∪ {±∞} is defined by g(y) = inf{K(x, y) | x ∈ ZV }

(y ∈ ZU ).

(8.60)

We say that the problem D is feasible if g(y) > −∞ for some y ∈ ZU . If the problem P is feasible, we have g(y) < +∞ for all y ∈ ZU , since g(y) ≤ K(x, y) ≤ f (x) for all x ∈ ZV and y ∈ ZU .

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We use the following notations: inf(P) = inf{f (x) | x ∈ ZV }, sup(D) = sup{g(y) | y ∈ ZU }, opt(P) = {x ∈ ZV | f (x) = inf(P)}, opt(D) = {y ∈ ZU | g(y) = sup(D)}. We write min(P) instead of inf(P) if the problem P is feasible and the infimum is finite, in which case the infimum is attained (i.e., opt(P) = ∅), and similarly for max(D). Theorem 8.52 (Weak duality). inf(P) ≥ sup(D). Proof. We have g(y) = inf K(x, y) = inf inf (F (x, u) + u, y ) x

x

u

≤ inf (F (x, 0) + 0, y ) = inf f (x) = inf(P). x

x

Hence, sup(D) = supy g(y) ≤ inf(P). Our main interest lies in the strong duality, namely, in the case where the inequality in the weak duality turns to an equality with a finite common value. Theorem 8.53. (1) g(y) = −ϕ• (−y). (2) sup(D) = ϕ•• (0). (3) inf(P) = ϕ(0). (4) inf(P) = sup(D) ⇐⇒ ϕ(0) = ϕ•• (0). (5) Suppose inf(P) is finite. Then min(P) = max(D) ⇐⇒ ∂Z ϕ(0) = ∅. (6) If min(P) = max(D), then opt(D) = −∂Z ϕ(0). Proof. (1) By the definitions we have −ϕ• (−y) = − sup( u, −y − ϕ(u)) = inf (ϕ(u) + u, y ) u

u

= inf (inf F (x, u) + u, y ) = inf inf (F (x, u) + u, y ) u

x

x

u

= inf K(x, y) = g(y). x

(2) By using (1) we have sup(D) = sup g(y) = sup( 0, −y − ϕ• (−y)) = sup( 0, y − ϕ• (y)) = ϕ•• (0). y

y

y

(3) This is obvious from (8.54) and (8.57). (4) The equivalence is due to (2) and (3). (5), (6) We have the following chain of equivalence: y ∈ −∂Z ϕ(0) ⇔ ϕ(u) − ϕ(0) ≥ u, −y (∀ u ∈ ZU ) ⇔ inf u (ϕ(u) + u, y ) = ϕ(0) ⇔ g(y) = ϕ(0), where

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inf u (ϕ(u) + u, y ) = g(y) is shown in the proof of (1). This implies the claims when combined with the weak duality (Theorem 8.52).

Theorem 8.54 (Saddle-point theorem). Both inf(P) and sup(D) are finite and min(P) = max(D) if and only if there exist x ∈ ZV and y ∈ ZU such that K(x, y) is finite and K(x, y) ≤ K(x, y) ≤ K(x, y)

(x ∈ ZV , y ∈ ZU ).

If this is the case, we have x ∈ opt(P) and y ∈ opt(D). Proof. By Proposition 8.51 (2) we have f (x) = supy K(x, y) for any x ∈ ZV , whereas g(y) = inf x K(x, y) for any y ∈ ZU by the definition (8.60). In view of the weak duality (Theorem 8.52) and the relation f (x) = sup K(x, y) ≥ K(x, y) ≥ inf K(x, y) = g(y), x

y

we see that min(P) = max(D) ⇐⇒ ∃ x, ∃ y : f (x) = g(y) ⇐⇒ ∃ x, ∃ y : sup K(x, y) = K(x, y) = inf K(x, y). y

8.4.3

x

Lagrangian Function Based on M-Convexity

As the perturbation F , we choose F = Fr : ZV × ZV → Z ∪ {+∞} defined by Fr (x, u) = c(x) + δB (x + u) + r(u)

(x, u ∈ ZV ),

(8.61)

where r : ZV → Z ∪ {+∞} is an M-convex function with r(0) = 0. (We take V as the U in the general framework.) The special case with r = 0 is distinguished by the subscript 0. Namely, F0 (x, u) = c(x) + δB (x + u)

(x, u ∈ ZV ).

(8.62)

We single out the case of r = 0 because the technical development in this special case can be made within the framework of M-/L-convex functions, whereas the general case involves M2 -/L2 -convex functions. Throughout this section we assume (REG), i.e., that B is an M-convex set. We use the subscript r to denote the quantities derived from Fr ; namely, (u ∈ ZV ), ϕr (u) = inf{Fr (x, u) | x ∈ ZV } (x, y ∈ ZV ), Kr (x, y) = inf{Fr (x, u) + u, y | u ∈ ZV }

(8.63) (8.64)

gr (y) = inf{Kr (x, y) | x ∈ ZV } sup(Dr ) = sup{gr (y) | y ∈ ZV }.

(8.65)

(y ∈ ZV ),

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Our choice of the perturbation (8.61) is legitimate, meeting the requirements (8.54) and (8.55), as follows. Proposition 8.55. Assume (REG). (1) Fr (x, 0) = f (x) (x ∈ ZV ). (2) For each x ∈ ZV , F0 (x, u) is M-convex in u or F0 (x, u) = +∞ for all u. (3) For each x ∈ ZV , Fr (x, u) is M2 -convex in u or Fr (x, u) = +∞ for all u. (4) Fr (x, ·)•• = Fr (x, ·) (x ∈ ZV ). (5) Fr (x, u) = sup{Kr (x, y) − u, y | y ∈ ZV } (x, u ∈ ZV ). Assume (REG) and (OBJ). (6) For each u ∈ ZV , Fr (x, u) is M2 -convex in x or Fr (x, u) = +∞ for all x. Proof. (1) This follows from r(0) = 0. (2) We have F0 (x, u) = +∞ unless x ∈ dom c. For each x, δB (x+u) = δB−x (u) is the indicator function of B−x (translation of B by x), which is again an M-convex set. Therefore, δB (x + u) is M-convex in u. (3) We have Fr (x, u) = +∞ unless x ∈ dom c. Besides δB (x + u), r(u) is M-convex by the assumption. Hence, Fr (x, ·) is the sum of two M-convex functions for each x ∈ dom c. By definition, such a function is either M2 -convex or identically equal to +∞. (4) This follows from (3) and Theorem 8.36. (5) This follows from (4) and Proposition 8.51 (1). (6) The proof is similar to (3) by the symmetry between c(x) and r(u). The Lagrangian function Kr (x, y) has the following properties. It should be clear that δB • is the support function of B and δ−B 2Z r[y] means the integer infimal convolution of the indicator function of −B = {x | −x ∈ B} and r[y](u) = r(u) + u, y . Proposition 8.56. (1)  c(x) − x, y − δB • (−y) (x ∈ dom c, y ∈ ZV ), K0 (x, y) = +∞ (x ∈ / dom c, y ∈ ZV ). ⎧ ⎨ c(x) + (δ−B 2Z r[y])(−x) = c(x) − (δB−x + r)• (−y) (x ∈ dom c, y ∈ ZV ), Kr (x, y) = ⎩ +∞ (x ∈ / dom c, y ∈ ZV ).

(2)

Proof. It suffices to prove (2), since (1) is its special case with r = 0. Assume x ∈ dom c. Substituting (8.61) into (8.64) we obtain Kr (x, y) = inf{c(x) + δB (x + u) + r(u) + u, y | u ∈ ZV } = c(x) + inf{δ−B (−x − u) + r[y](u) | u ∈ ZV } = c(x) + (δ−B 2Z r[y])(−x). The alternative expression is easy to see.

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Theorem 8.57. Assume (REG). (1) For each x ∈ ZV , K0 (x, y) is L-concave in y or K0 (x, y) = +∞ for all y. (2) For each x ∈ ZV , Kr (x, y) is L2 -concave in y or Kr (x, y) = +∞ for all y. Assume (REG) and (OBJ). (3) For each y ∈ ZV , K0 (x, y) is M-convex in x or K0 (x, y) ∈ {+∞, −∞} for all x. (4) For each y ∈ ZV , Kr (x, y) is M2 -convex in x or Kr (x, y) ∈ {+∞, −∞} for all x. Proof. (1), (3) The expression of K0 (x, y) in Proposition 8.56 (1) shows these. (2) In the expression of Kr (x, y) in Proposition 8.56 (2) we have δB−x + r ∈ M2 [Z → Z] or ≡ +∞ (see Note 6.17). Then the conjugacy in Theorem 8.48 implies this. (4) In the expression of Kr (x, y) in Proposition 8.56 (2) the second term (δ−B 2Z r[y])(−x) is M-convex or ∈ {+∞, −∞} since it is the integer infimal convolution of two M-convex functions (Theorem 6.13 (8)). In the case of M-convex programs the dual objective function gr and the optimal value function ϕr are well behaved, as follows. Theorem 8.58. Assume (REG) and (OBJ). (1) g0 is L-concave or g0 (y) = −∞ for all y. (2) gr is L2 -concave, gr (y) = −∞ for all y, or gr (y) = +∞ for all y. (3) ϕ0 is M-convex or ϕ0 (u) ∈ {+∞, −∞} for all u. (4) ϕr is M2 -convex or ϕr (u) ∈ {+∞, −∞} for all u. Proof. We prove (1), (3), (4), and, finally, (2). (1) Using Proposition 8.56 (1) we obtain g0 (y) = inf K0 (x, y) = inf (c(x) − x, y ) − δB • (−y) = −c• (y) − δB • (−y). x

x

This shows g0 is L-concave or ≡ −∞, since the sum of two L-concave functions is again L-concave provided the effective domains of the summands are not disjoint. (3) We have ϕ0 (u) = inf x (c(x) + δB (x + u)) = (c2Z δ−B )(−u). The assertion follows from Theorem 6.13 (8). (4) It follows from Fr (x, u) = F0 (x, u) + r(u) that ϕr (u) = +∞ unless u ∈ dom r and that ϕr (u) = ϕ0 (u) + r(u) if u ∈ dom r. If ϕ0 ∈ M[Z → Z], then ϕr ∈ M2 [Z → Z] or ϕr ≡ +∞. If ϕ0 (u) ∈ {+∞, −∞}, then ϕr (u) ∈ {+∞, −∞}. (2) First recall the relation gr (y) = −ϕr • (−y) (Theorem 8.53 (1)). If ϕr ∈ M2 [Z → Z], the conjugacy in Theorem 8.48 implies the L2 -concavity of gr . If ϕr ≡ +∞, then gr ≡ +∞. If ϕr (u) = −∞ for some u, then gr ≡ −∞. Strong duality holds true for M-convex programs with the Lagrangian function Kr (x, y). Theorem 8.59 (Strong duality). Assume (REG), (OBJ), and that the problem P

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is feasible and bounded from below. (1) min(P) = ϕr (0) = ϕr •• (0) = max(Dr ). (2) opt(Dr ) = −∂Z ϕr (0). Proof. Since ϕr (0) is finite by the assumption, ϕr is M2 -convex by Theorem 8.58 (4) and ∂Z ϕr (0) = ∅ by Theorem 8.35. Then the assertions follow from Theorem 8.53. It should be emphasized that the M-convexity of the objective function c is a sufficient condition and not an absolute prerequisite for the strong duality to hold. Example 8.60. Let us consider the case where c(x) is a linear function on another M-convex set B  ⊆ ZV . The primal problem with c(x) = x, w + δB  (x) (where w ∈ ZV denotes a weight vector) reads as follows: P:

Minimize x, w

subject to x ∈ B ∩ B  .

The Lagrangian function K0 is given by  x, w − y + inf z∈B z, y (x ∈ B  , y ∈ ZV ), K0 (x, y) = +∞ (x ∈ / B  , y ∈ ZV ), from which is derived the following dual problem:   D: Maximize inf  x, w − y + inf z, y

x∈B

z∈B

subject to

y ∈ ZV .

This is the polymatroidal version of the optimal common base problem explained in Example 8.27. The optimal solution y = y ∗ to D gives the weight splitting w1∗ = w − y ∗ and w2∗ = y ∗ . For a concrete instance, take V = {1, 2}, B = {x ∈ Z2 | x(1) ≥ 0, x(1) + x(2) = 0}, B  = {x ∈ Z2 | x(1) ≤ 1, x(1) + x(2) = 0}. We have B ∩ B  = {(0, 0), (1, −1)} and ⎧ ⎨ x, w − y (x ∈ B  , y(1) ≥ y(2)), −∞ (x ∈ B  , y(1) < y(2)), K0 (x, y) = ⎩ +∞ (otherwise).

8.4.4

(8.66)

Symmetry in Duality

So far we have derived the dual problem D from the primal P by means of a perturbation function F (x, u) such that F (x, 0) = f (x) and F (x, ·) ∈ M2 [Z → Z]. Namely, P : minimize f (x)

−−−−−−−−−−−−−−→ F (x, 0) = f (x) F (x, ·) ∈ M2 [Z → Z]

D : maximize g(y).

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We have seen that g is L2 -concave, i.e., −g ∈ L2 [Z → Z], in the “convex” case where (REG) and (OBJ) are satisfied. We are now interested in the reverse process, i.e., how to restore the primal problem P from the dual D in a way consistent with the general duality framework of section 8.4.1. We embed the dual problem D in a family of maximization problems defined in terms of another perturbation function G(y, v) such that G(y, 0) = g(y) and −G(y, ·) ∈ L2 [Z → Z]. Namely, P : minimize f (x)

←−−−−−−−−−−−−−− G(y, 0) = g(y) −G(y, ·) ∈ L2 [Z → Z]

D : maximize g(y).

With reference to (8.60) and Proposition 8.51 we define a perturbation function G : ZU × ZV → Z ∪ {±∞} by54 G(y, v) = inf{K(x, y) − x, v | x ∈ ZV }

(y ∈ ZU , v ∈ ZV ).

(8.67)

By this we intend to consider a family of maximization problems parametrized by v ∈ ZV : Maximize G(y, v) subject to y ∈ ZU . The optimal value function γ : ZV → Z ∪ {±∞} is accordingly defined by γ(v) = sup{G(y, v) | y ∈ ZU }

(v ∈ ZV ).

(8.68)

˜ It is then natural to introduce the dual Lagrangian function K(x, y) : ZV × ZU → Z ∪ {±∞} as ˜ K(x, y) = sup{G(y, v) + x, v | v ∈ ZV }

(x ∈ ZV , y ∈ ZU ).

(8.69)

The problem dual to the problem D is to minimize ˜ f˜(x) = sup{K(x, y) | y ∈ ZU }

(x ∈ ZV ).

(8.70)

As can be imagined from the corresponding constructions in convex analysis (cf. sec˜ tion 4 of Rockafellar [177]), K(x, y) and f˜(x) thus constructed do not necessarily coincide with the original K(x, y) and f (x). We show, however, that the dual of the dual comes back to the primal in the canonical case with a bounded M-convex set B using the Lagrangian function Kr . Example 8.61. For K0 of (8.66) in Example 8.60 we can calculate ⎧ ⎨ x, w − y (x ∈ B  , y(1) ≥ y(2)), ˜ +∞ (x ∈ / B  , y(1) ≥ y(2)), K0 (x, y) = ⎩ −∞ (y(1) < y(2)). ˜ 0 (x, y) = K0 (x, y) where they take finite values. We observe that K 54 Here

we have v ∈ ZV and not v ∈ V .

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In what follows we always assume (REG), (OBJ), and that B is bounded. We consider the Lagrangian function Kr with U = V . Proposition 8.62. Assume (REG), (OBJ), and that B is bounded. Then f˜(x) = f (x)

(x, y ∈ ZV ),

˜ r (x, y) = Kr (x, y) K

(x ∈ ZV ).

˜ r (·, y) = Proof. The definitions (8.67) and (8.69) show Gr (y, ·) = −(Kr (·, y))• and K • (−Gr (y, ·)) for each y. Since Kr (·, y) ∈ M2 [Z → Z] or ≡ +∞ by Theorem 8.57 (4) when B is bounded, we have Kr (·, y) = (Kr (·, y))•• for each y. Hence follows ˜ r = Kr . Then Proposition 8.51 (2) and (8.70) imply f˜ = f . K

Proposition 8.63. Assume (REG), (OBJ), and that B is bounded. (1) Gr (y, 0) = gr (y) (y ∈ ZV ). (2) For each y ∈ ZV , Gr (y, v) is L2 -concave in v or Gr (y, v) = +∞ for all v. (3) For each v ∈ ZV , Gr (y, v) is L2 -concave in y, Gr (y, v) = −∞ for all y, or Gr (y, v) = +∞ for all y. Proof. (1) This is obvious from (8.65) and (8.67). (2) The definition (8.67) shows Gr (y, ·) = −(Kr (·, y))• for each y, while Kr (·, y) ∈ M2 [Z → Z] or ≡ +∞ by Theorem 8.57 (4) when B is bounded. Hence −Gr (y, ·) ∈ L2 [Z → Z] or ≡ −∞ by Theorem 8.48. (3) By (8.67), (8.64), and (8.61), we have the expression Gr (y, v) = inf inf (Fr (x, u) + u, y − x, v ) x

u

= inf inf (c[−v](x) + δB (x + u) + r(u) + u, y ), x

u

in which c[−v] is M-convex. On the other hand, Theorem 8.58 (2) shows that gr (y) = inf inf (c(x) + δB (x + u) + r(u) + u, y ) x

u

is L2 -concave, gr (y) = −∞ for all y, or gr (y) = +∞ for all y. By replacing c with c[−v] we obtain the claim. The optimal value function γr , defined by (8.68) with reference to Gr , enjoys the following properties. Theorem 8.64. Assume (REG), (OBJ), and that B is bounded. (1) f (x) = −γr ◦ (−x). (2) γr is L2 -concave or γr (v) = +∞ for all v. Proof. (1) Using Proposition 8.51 (2), (8.69), and Proposition 8.62 we obtain f (x) = sup Kr (x, y) = sup sup(Gr (y, v) + x, v ) y

y

v

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Chapter 8. Conjugacy and Duality   = sup (sup Gr (y, v)) + x, v

v

y

= sup(γr (v) + x, v ) = −γr ◦ (−x). v

(2) Since f ∈ M2 [Z → Z] or ≡ +∞, the assertion follows from (1) and the conjugacy between L2 [Z → Z] and M2 [Z → Z].

Theorem 8.65. Assume (REG), (OBJ), and that the problem P is feasible and B is bounded. (1) min(P) = γr (0) = γr ◦◦ (0) = max(Dr ). (2) opt(P) = ∂Z (−γr )(0) = ∅. Proof. The proof is essentially the same as that of Theorem 8.59. To be specific, we have the following chain of equivalence: x ∈ ∂Z (−γr )(0) ⇔ γr (v) − γr (0) ≤ v, −x (∀ v ∈ ZV ) ⇔ − inf v ( v, −x − γr (v)) = γr (0) ⇔ f (x) = γr (0). This implies the claim when combined with the weak duality (Theorem 8.52).

Bibliographical Notes The conjugacy relationship between M-convexity and L-convexity was established first for integer-valued functions (Theorem 8.12) by Murota [140], whereas the present proof is based on Murota [147]. The conjugacy theorem for polyhedral M-/L-convex functions (Theorem 8.4) is due to Murota–Shioura [152]. The polarity between M-/L-convex cones in Theorem 8.5 is stated in [147] and Proposition 8.11 for the integer biconjugate is in [140]. Theorem 8.1 is a special case of a theorem of Topkis [202], stated explicitly as Corollary 2.7.3 in Topkis [203]. The M-separation theorem (Theorem 8.15) is given in Murota [137], [140], [142] and the L-separation theorem (Theorem 8.16) in [140]. The Fenchel-type duality theorem for M-convex functions originated in [137] (see also [140]); the present form (Theorem 8.21) is in Murota [147]. The M-convex intersection theorem (Theorem 8.17) is in [137], [142]. The Fenchel-type duality theorem for submodular set functions described in Example 8.26 is due to Fujishige [62]. The weight-splitting theorem for weighted matroid intersection in Example 8.27 is due to Frank [54], and that for valuated matroid intersection in Example 8.28 is due to Murota [135]; see also Theorem 5.2.40 of Murota [146]. M2 -convex and L2 -convex functions were introduced by Murota [140], to which Theorems 8.35, 8.36, 8.45, and 8.46 and the conjugacy theorem (Theorem 8.48) are ascribed. Theorems 8.31 and 8.42 (integral convexity) as well as Theorem 8.49 are due to Murota–Shioura [153]. Theorems 8.32 and 8.43 (M2 -/L2 -optimality criteria) and Theorems 8.34 and 8.44 (M2 -/L2 -proximity theorems) are given by Murota– Tamura [162]. See Tamura [198] for Notes 8.37 and 8.38. The Lagrange duality of section 8.4 is developed in Murota [140]. See Nemhauser– Rinnooy Kan–Todd [166] and Nemhauser–Wolsey [167] for the subadditive duality.

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Chapter 9

Network Flows

In Chapter 2 we had a glimpse of the intrinsic relationship between M-/L-convexity and network flows (nonlinear electrical networks). Pursuing this direction further we show the following facts in this chapter. (i) The minimum cost flow problem can be generalized to the submodular flow problem, where M-/L-convexity plays a fundamental role. (ii) The submodular flow problem with an M-convex function admits nice optimality criteria in terms of potentials and negative cycles. (iii) The optimality criterion using potentials is equivalent to the Fenchel-type duality theorem. (iv) A conjugate pair of M-convex and L-convex functions is transformed to another conjugate pair of M-convex and L-convex functions through network flows. Algorithms are treated in Chapter 10.

9.1

Minimum Cost Flow and Fenchel Duality

To single out the role of M-/L-convexity we first review standard results on the conventional minimum cost flow problem. Emphasis is placed on the equivalence of the optimality criterion in terms of potentials and the Fenchel duality theorem for convex functions.

9.1.1

Minimum Cost Flow Problem

Let G = (V, A) be a directed graph with vertex set V and arc set A. Suppose that each arc a ∈ A is associated with an upper capacity c(a), a lower capacity c(a), and a cost γ(a) per unit flow. Furthermore, for each vertex v ∈ V , the amount of flow supply at v is specified by x(v). The minimum cost flow problem is to find a flow ξ = (ξ(a) | a ∈ A) that minimizes the total cost γ, ξ A = a∈A γ(a)ξ(a) subject to the capacity constraint and the supply specification. Here the supply specification means a constraint that the boundary ∂ξ of ξ, defined by ∂ξ(v) =



{ξ(a) | a ∈ δ + v} −



245

{ξ(a) | a ∈ δ − v}

(v ∈ V ),

(9.1)

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should be equal to the given x. The problem is described by a graph G = (V, A), an upper capacity c : A → R ∪ {+∞}, a lower capacity c : A → R ∪ {−∞}, a cost vector γ : A → R, and a supply vector x : V → R, where it is assumed that c(a) ≥ c(a) for each a ∈ A. The variable to be optimized is the flow ξ : A → R. Minimum cost flow problem MCFP0 (linear arc cost)55  Minimize Γ1 (ξ) = γ(a)ξ(a)

(9.2)

a∈A

subject to c(a) ≤ ξ(a) ≤ c(a) ∂ξ = x, ξ(a) ∈ R

(a ∈ A),

(a ∈ A).

(9.3) (9.4) (9.5)

The minimum cost flow problem is a typical well-behaved combinatorial problem that has nice properties, such as 1. an optimality criterion in terms of potentials (dual variables), 2. an optimality criterion in terms of negative cycles, 3. the integrality of optimal solutions, and 4. efficient algorithms. Precise statements for the first three above are given later in Theorems 9.4, 9.5, and 9.6, respectively. In particular, the integrality of optimal solutions refers to the fact that, if the capacity constraint and the supply specification are given in terms of integer-valued functions, c : A → Z ∪ {+∞}, c : A → Z ∪ {−∞}, and x : V → Z, then there exists an integer-valued optimal flow ξ to the above problem. This implies that the problem MCFP0 specified by such integer-valued data remains essentially the same even if the integrality condition ξ(a) ∈ Z

(a ∈ A)

(9.6)

is additionally imposed on the flow ξ. We refer to the problem with (9.6) in place of (9.5) as the minimum cost integer-flow problem. To discuss the relationship to convex analysis it is convenient to consider a more general form of the minimum cost flow problem. The generalization is twofold. is replaced with a nonlinear cost represented First, the linear arc cost a∈A γ(a)ξ(a) by a separable convex function a∈A fa (ξ(a)) with a family of univariate polyhedral convex functions fa ∈ C[R → R] indexed by a ∈ A. Second, with a polyhedral convex function f : RV → R ∪ {+∞}, an additional term f (∂ξ) for the flow boundary ∂ξ is introduced in the cost function as a generalization of the supply specification ∂ξ = x. Minimum cost flow problem MCFP3 (nonlinear cost)56  Minimize Γ3 (ξ) = fa (ξ(a)) + f (∂ξ)

(9.7)

a∈A

subject to ξ(a) ∈ dom fa 55 MCFP 56 We

(a ∈ A),

(9.8)

stands for minimum cost flow problem. have MCFPi for i = 0, 3 and not for i = 1, 2. This is for consistency with section 9.2.

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247

∂ξ ∈ dom f,

(9.9)

ξ(a) ∈ R

(a ∈ A).

(9.10)

Obviously, MCFP0 is a special case of MCFP3 , where  γ(a)t (t ∈ [c(a), c(a)]), fa (t) = +∞ (otherwise)

(9.11)

for a ∈ A and f is the indicator function δ{x} of the singleton set {x}. Among the four nice properties of MCFP0 listed above, the optimality criterion by potentials is generalized to MCFP3 , as we will see in section 9.1.3, whereas the other three fail to survive for a general f . In considering the integer-flow version of the problem it is natural to assume fa ∈ C[Z → R] (or fa ∈ C[Z|R → R]) for each a ∈ A, but it is not clear what combinatorial property to impose on f to ensure the integrality of optimal solutions. M-convexity gives an answer to this, as we will see in section 9.4. Note 9.1. In MCFP3 we have restricted f and fa (a ∈ A) to be polyhedral convex functions. This is for consistency with our theoretical framework of polyhedral M-/L-convex functions. The optimality criterion by potentials (Theorem 9.4), as well as its equivalence to the Fenchel duality to be discussed in section 9.1.4, remains valid for nonpolyhedral convex functions under appropriate assumptions; see Iri [94] and Rockafellar [178].

9.1.2

Feasibility

For the minimum cost flow problem MCFP0 , a feasible flow means a function ξ : A → R that satisfies c(a) ≤ ξ(a) ≤ c(a)

(a ∈ A),

(9.12)

∂ξ = x.

(9.13)

We say that MCFP0 is feasible if it admits a feasible flow. For X ⊆ V we denote the sets of arcs leaving and entering X by Δ+ X = {a ∈ A | ∂ + a ∈ X, ∂ − a ∈ V \ X}, −



(9.14)

Δ X = {a ∈ A | ∂ a ∈ X, ∂ a ∈ V \ X} +

and define the cut capacity function κ : 2V → R ∪ {+∞} by   c(a) − c(a) κ(X) = c(Δ+ X) − c(Δ− X) = a∈Δ+ X

(9.15)

(X ⊆ V ).

a∈Δ− X

Proposition 9.2. The cut capacity function κ is submodular. Proof. It is easy to verify κ(X) + κ(Y ) − κ(X ∪ Y ) − κ(X ∩ Y ) =

 a

[ c(a) − c(a) ] ≥ 0,

(9.16)

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where the summation is taken over all arcs a connecting X \ Y and Y \ X. If a flow ξ meets the capacity constraint (9.12), its boundary x = ∂ξ satisfies     x(X) = ∂ξ(X) = ξ(a) − ξ(a) ≤ c(a) − c(a) = κ(X) a∈Δ+ X

a∈Δ− X

a∈Δ+ X

a∈Δ− X

for all X ⊆ V and also x(V ) = 0 = κ(V ). This means x ∈ B(κ), where B(κ) is the base polyhedron (4.13) associated with κ. The above argument shows that the condition x ∈ B(κ) is necessary for MCFP0 to be feasible. It is also sufficient, as stated in the following theorem. Theorem 9.3 (Feasibility). For c : A → R ∪ {+∞}, c : A → R ∪ {−∞}, and x : V → R, there exists a flow ξ : A → R satisfying (9.12) and (9.13) if and only if x(X) ≤ κ(X)

(∀ X ⊆ V ),

x(V ) = 0.

(9.17)

That is, B(κ) = {∂ξ | ξ : A → R, c(a) ≤ ξ(a) ≤ c(a) (a ∈ A)}.

(9.18)

If c and c are integer valued, we may restrict ξ to be integer flows; namely, B(κ) ∩ ZV = {∂ξ | ξ : A → Z, c(a) ≤ ξ(a) ≤ c(a) (a ∈ A)}.

(9.19)

Proof. This follows from the max-flow min-cut theorem or a variant thereof, called Hoffman’s circulation theorem (see, e.g., (2.65) of Fujishige [65] or Theorem 3.18 of Cook–Cunningham–Pulleyblank–Schrijver [26]).

9.1.3

Optimality Criteria

The minimum cost flow problem MCFP3 , which has convex boundary cost and separable convex arc cost, admits a nice optimality criterion in terms of potentials. The conventional case MCFP0 admits, in addition, an optimality criterion in terms of negative cycles and the integrality of optimal solutions. A potential means a function p : V → R (or a vector p ∈ RV ) on the vertex set. The coboundary of a potential p is a function δp : A → R defined by δp(a) = p(∂ + a) − p(∂ − a)

(a ∈ A).

(9.20)

The inner product (pairing) of tension η : A → R and flow ξ : A → R can be expressed as (9.21) η, ξ A = − δp, ξ A = − p, ∂ξ V = − p, x V if x = ∂ξ and p is a potential such that η(a) = −δp(a) = p(∂ − a) − p(∂ + a)

(a ∈ A).

(9.22)

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The identity (9.21) is a fundamental relation, frequently used in the subsequent arguments. It should be clear that   η, ξ A = η(a)ξ(a), p, x V = p(v)x(v). a∈A

v∈V

With reference to a potential p we modify the cost functions f and fa (a ∈ A) to the reduced cost functions f [−p] and fa [δp(a)] (a ∈ A) defined by  f [−p](x) = f (x) − p(v)x(v) (x ∈ RV ), (9.23) v∈V

fa [δp(a)](t) = fa (t) + (p(∂ + a) − p(∂ − a))t

(t ∈ R).

(9.24)

A straightforward calculation with the use of (9.21) yields  fa (ξ(a)) + f (∂ξ) Γ3 (ξ) = a∈A

8 =



9 fa (ξ(a)) + δp, ξ A

+ (f (∂ξ) − p, ∂ξ V )

a∈A

=



fa [δp(a)](ξ(a)) + f [−p](∂ξ)

a∈A





inf fa [δp(a)] + inf f [−p],

(9.25)

a∈A

where inf f [−p] and inf fa [δp(a)] with a ∈ A mean the infima of the reduced cost functions. The inequality (9.25) gives a lower bound for the minimum of Γ3 . In particular, if (i) ξ(a) ∈ arg min fa [δp(a)] for every a ∈ A and (ii) ∂ξ ∈ arg min f [−p] for some p, then ξ is an optimal flow satisfying (9.25) with equality. This statement is true for any functions f and fa (a ∈ A). The converse is also true under a fairly general assumption that f and fa (a ∈ A) are convex. Theorem 9.4 (Potential criterion). In the minimum cost flow problem MCFP3 with polyhedral convex f and fa (a ∈ A), we have the following: (1) For a feasible flow ξ : A → R, the two conditions (OPT) and (POT) below are equivalent. (OPT) ξ is an optimal flow. (POT) There exists a potential p : V → R such that (i) ξ(a) ∈ arg min fa [δp(a)] for every a ∈ A, and (ii) ∂ξ ∈ arg min f [−p]. (2) Suppose that a potential p : V → R satisfies (i) and (ii) above for an optimal flow ξ. A feasible flow ξ  is optimal if and only if

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Chapter 9. Network Flows (i) ξ  (a) ∈ arg min fa [δp(a)] for every a ∈ A, and (ii) ∂ξ  ∈ arg min f [−p].

Proof. (1) (POT) ⇒ (OPT) is already shown. To prove (OPT) ⇒ (POT), suppose that ξ is an optimal flow. Putting ) * ( )  )   fA (x) = inf fa (ξ (a))) ∂ξ = x , (9.26) ) ξ a∈A

we see

+8 

inf Γ3 (ξ ) = inf ξ

x

inf

ξ  :∂ξ  =x



9 

fa (ξ (a))

, + f (x) = inf [fA (x) + f (x)], x

a∈A

where inf Γ3 is finite and x = ∂ξ attains the infimum of the last expression. Noting that fA is a polyhedral convex function (see Note 2.17) and dom fA ∩ dom f = ∅, we apply the Fenchel duality theorem in convex analysis (Theorem 3.6 and (3.42)) to obtain p : V → R such that fA [p](∂ξ) = inf fA [p],

f [−p](∂ξ) = inf f [−p].

(9.27)

The second equation shows (ii) in (POT). We will show that the first equation above implies (i) in (POT). It follows from (9.26) and (9.21) that ) ( * )  ) fA [p](x ) = inf fa (ξ  (a))) ∂ξ  = x + p, x V ) ξ a∈A ) ( * )  )     = inf fa (ξ (a)) + δp, ξ A ) ∂ξ = x ) ξ a∈A ) ( * )  )    = inf fa [δp(a)](ξ (a))) ∂ξ = x ) ξ a∈A

for any x ∈ RV , and therefore, inf fA [p] =



inf fa [δp(a)].

(9.28)

a∈A

On the other hand, the optimality of ξ implies fA (∂ξ) = combination with (9.21), yields  fa [δp(a)](ξ(a)). fA [p](∂ξ) =

a∈A

fa (ξ(a)), which, in

a∈A

Substituting (9.28) and (9.29) into the first equation in (9.27) shows fa [δp(a)](ξ(a)) = inf fa [δp(a)]

(∀ a ∈ A),

(9.29)

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251

η(a) = 6 p(∂ − a) − p(∂ + a) Γa 0

γp (a)

γ(a)

? c(a)

c(a) ξ(a)

Figure 9.1. Characteristic curve (kilter diagram) for linear cost .

which is equivalent to (i) in (POT). (2) This is obvious from (1) and (9.25). A potential p satisfying (i) and (ii) in (POT) is called an optimal potential . Though this definition refers to a particular optimal flow ξ, it is, in fact, independent of the choice of ξ by Theorem 9.4 (2). Condition (i) in (POT) is closely related to the characteristic curve (or kilter diagram) Γa introduced in section 2.2 with an illustration in Fig. 2.3. Since ξ(a) ∈ arg min fa [−η(a)] ⇐⇒ (ξ(a), η(a)) ∈ Γa

(9.30)

by (2.34) and (2.35), condition (i) in (POT) says that flow ξ(a) and tension η(a) = −δp(a) should satisfy the constitutive equation in every arc a ∈ A. In the case of linear arc cost, the characteristic curve Γa takes the form of Fig. 9.1, and, accordingly, condition (i) in (POT) is expressed as γp (a) > 0 =⇒ ξ(a) = c(a),

(9.31)

γp (a) < 0 =⇒ ξ(a) = c(a)

(9.32)

in terms of the reduced cost γp : A → R defined by γp (a) = γ(a) + p(∂ + a) − p(∂ − a)

(a ∈ A).

(9.33)

In the conventional case MCFP0 with linear arc cost, the optimality criterion can be reformulated in terms of negative cycles in an auxiliary network. For a feasible flow ξ : A → R, let Gξ = (V, Aξ ) be a directed graph with vertex set V and arc set Aξ = A∗ξ ∪ Bξ∗ consisting of two disjoint parts: A∗ξ = {a | a ∈ A, ξ(a) < c(a)},

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Chapter 9. Network Flows Bξ∗ = {a | a ∈ A, c(a) < ξ(a)}

(a: reorientation of a),

and define a function ξ : Aξ → R, representing arc lengths, by  γ(a) (a ∈ A∗ξ ), ξ (a) = −γ(a) (a ∈ Bξ∗ , a ∈ A).

(9.34)

We refer to (Gξ , ξ ) as the auxiliary network . We call a directed cycle of negative length a negative cycle. Theorem 9.5 (Negative-cycle criterion). For a feasible flow ξ : A → R to the minimum cost flow problem MCFP0 , conditions (OPT) and (NNC) below are equivalent. (OPT) ξ is an optimal flow. (NNC) There exists no negative cycle in (Gξ , ξ ) with ξ of (9.34). Proof. By (9.31), (9.32), and the definition (9.34) of ξ , condition (i) of (POT) in Theorem 9.4 is equivalent to ξ (a) + p(∂ + a) − p(∂ − a) ≥ 0

(a ∈ Aξ ),

(9.35)

whereas condition (ii) of (POT) is void for MCFP0 . On the other hand, the existence of a potential p : V → R satisfying (9.35) is equivalent to (NNC), as is well known in network flow theory. Hence follows the equivalence of (NNC) and (OPT) by Theorem 9.4. The minimum cost flow problem MCFP0 is endowed with remarkable integrality properties: 1. An integer-valued optimal flow exists if the upper and lower capacities and the supply vector are integer valued (primal integrality). 2. An integer-valued optimal potential exists if the cost vector is integer valued (dual integrality). Theorem 9.6 (Integrality). Suppose that the minimum cost flow problem MCFP0 has an optimal solution. (1) [Primal integrality] If c : A → Z ∪ {+∞}, c : A → Z ∪ {−∞}, and x : V → Z, then there exists an integer-valued optimal flow ξ : A → Z. (2) [Dual integrality] The set of optimal potentials Π∗ = {p | p : optimal potential } is an L-convex polyhedron. If γ : A → Z, then Π∗ is an integral L-convex polyhedron and there exists an integer-valued optimal potential p : V → Z. Proof. (1) Let p be an optimal potential. By (9.31) and (9.32), a flow ξ is optimal if and only if it is a feasible flow with respect to a more restrictive capacity constraint c∗ (a) ≤ ξ(a) ≤ c∗ (a) with ⎧ ⎧ ⎨ c(a) (γp (a) > 0), ⎨ c(a) (γp (a) > 0), c(a) (γp (a) = 0), c(a) (γp (a) = 0), c∗ (a) = c∗ (a) = ⎩ ⎩ c(a) (γp (a) < 0), c(a) (γp (a) < 0)

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for each a ∈ A. Since c∗ (a) and c∗ (a) are integers for every a ∈ A, the claim follows from (9.19) in Theorem 9.3. (2) Since condition (i) of (POT) in Theorem 9.4 is equivalent to (9.35) in the proof of Theorem 9.5, Π∗ coincides with the polyhedron described by (9.35) with an optimal ξ. This implies the L-convexity of Π∗ (see section 5.6). The integrality assertion follows from Proposition 5.1 (4). The nice features of the minimum cost flow problem discussed so far (Theorems 9.4, 9.5, and 9.6) are derived mainly from the combinatorial structure inherent in the underlying graph, as well as the convexity of the cost functions. Further combinatorial properties stemming from the M-convexity of the cost functions will be investigated in section 9.4 and section 9.5. Note 9.7. Here is a comment on the definition of the coboundary. In this book we follow the convention of defining δp(a) by δp(a) = (p at the initial vertex of a) − (p at the terminal vertex of a). The boundary ∂ξ(v) is defined to be the amount of flow leaving v and the tension η is defined as η = −δp (see (9.1), (9.20), and (9.22)). Then follows the fundamental identity η, ξ A = − δp, ξ A = − p, ∂ξ V . Another convention of defining δp(a) by δp(a) = (p at the terminal vertex of a) − (p at the initial vertex of a) and the tension η by η = δp results in η, ξ A = δp, ξ A = − p, ∂ξ V . The notations div and Δ in Rockafellar [178] are related to ours as div = ∂ and Δ = −δ.

9.1.4

Relationship to Fenchel Duality

We discuss here the relationship between the potential criterion for optimality for the minimum cost flow problem MCFP3 and the Fenchel duality in convex analysis. The potential criterion for MCFP3 (Theorem 9.4 in section 9.1.3) has been derived from the Fenchel duality applied to f and −fA , where ) ( * )  ) fA (x) = inf fa (ξ(a))) ∂ξ = x ξ ) a∈A

and the evaluation of fA amounts to solving a minimum cost flow problem with nonlinear arc cost fa but without boundary cost f . Thus, the minimum cost flow problem MCFP3 with boundary cost can be understood as a composition of the

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254

Chapter 9. Network Flows -

V1 f1 (x1 )

V2 −h2 (−x2 )

Figure 9.2. Minimum cost flow problem for Fenchel duality.

minimization/maximization problem of the Fenchel duality and the minimum cost flow problem without boundary cost. The proof of Theorem 9.4 yields, as a byproduct, a min-max identity for MCFP3 : inf

ξ∈RA ,x∈RV

{Φ(ξ, x) | ∂ξ = x} =

sup η∈RA ,p∈RV

{Ψ(η, p) | η = −δp},

(9.36)

where Φ(ξ, x) = f (x) +



fa (ξ(a))

(ξ ∈ RA , x ∈ RV ),

a∈A



Ψ(η, p) = −g(p) −

ga (η(a))

(η ∈ RA , p ∈ RV ),

a∈A

with g = f • and ga = fa • for a ∈ A. The identity (9.36) is an immediate consequence of the Fenchel duality (3.41): inf [fA (x) + f (x)] = sup [−fA • (−p) − f • (p)] , x

p

in which f • (p) = g(p) and fA • (−p) =

 a∈A

fa • (−δp(a)) =



ga (−δp(a)),

a∈A

by (9.28). The left-hand side of (9.36) is MCFP3 in disguise, and accordingly, we may think of the maximization problem on the right-hand side of (9.36) as an optimization problem dual to MCFP3 . Although the potential criterion for MCFP3 has been derived from the Fenchel duality, they are essentially equivalent, which we demonstrate here. To be specific, we derive the Fenchel duality theorem (Theorem 3.6, Case (a2)) from the optimality criterion for MCFP3 (Theorem 9.4). Given a polyhedral convex function f1 : RV → R ∪ {+∞} and a polyhedral concave function h2 : RV → R ∪ {−∞} with dom f1 ∩ dom h2 = ∅, we consider a minimum cost flow problem MCFP3 on the bipartite graph G = (V1 ∪ V2 , A) in Fig. 9.2. The vertex set of G consists of two copies of V , i.e., V1 and V2 , and the

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255

arc set is A = {(v1 , v2 ) | v ∈ V }, with v1 ∈ V1 and v2 ∈ V2 denoting the copies of v ∈ V . We define the boundary cost function f : RV1 × RV2 → R ∪ {+∞} by f (x1 , x2 ) = f1 (x1 ) − h2 (−x2 )

(x1 ∈ RV1 , x2 ∈ RV2 )

and assume that the arc cost functions fa (a ∈ A) are identically zero without capacity constraints. Note that x1 = −x2 if (x1 , x2 ) = ∂ξ for a flow ξ in this network. Assuming inf(f1 − h2 ) > −∞, let ξ be an optimal flow, which exists since f is a polyhedral convex function. Let (p1 , p2 ) ∈ RV1 × RV2 be an optimal potential satisfying (POT) in Theorem 9.4. Condition (i) of (POT) implies p1 = p2 . Since f [−(p1 , p2 )](x1 , x2 ) = f1 [−p1 ](x1 ) − h2 [−p2 ](−x2 ), condition (ii) of (POT) gives x ∈ arg min f1 [−p] ∩ arg max h2 [−p] for x = ∂ξ|V1 and p = p1 . This implies the Fenchel duality (3.41) for f1 and h2 ; see also (3.30) and (3.42).

9.2

M-Convex Submodular Flow Problem

A series of generalizations of the minimum cost flow problem to the M-convex submodular flow problem is described. Recall the conventional minimum cost flow problem MCFP0 introduced in section 9.1.1. It is described by a graph G = (V, A), an upper capacity c : A → R ∪ {+∞}, a lower capacity c : A → R ∪ {−∞}, a cost vector γ : A → R, and a supply vector x : V → R, where c(a) ≥ c(a) for each a ∈ A. A generalization of MCFP0 is obtained by relaxing the supply specification ∂ξ = x to the constraint that ∂ξ belong to a given set B of feasible or admissible supplies: ∂ξ ∈ B. (9.37) The nice properties described in section 9.1 are maintained if B is a base polyhedron represented as B = B(ρ) with a submodular set function ρ : 2V → R ∪ {+∞}. Such a problem described by some ρ ∈ S[R] is called the submodular flow problem. Submodular flow problem MSFP1 (linear arc cost)57 Minimize

Γ1 (ξ) =



γ(a)ξ(a)

(9.38)

a∈A

subject to c(a) ≤ ξ(a) ≤ c(a) ∂ξ ∈ B(ρ), ξ(a) ∈ R

(a ∈ A).

(a ∈ A),

(9.39) (9.40) (9.41)

57 MSFP stands for M-convex submodular flow problem. We use the notation MSFP with i i = 1, 2, 3 to indicate the hierarchy of generality in the problems.

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In the integer-flow version of the problem, with ξ(a) ∈ Z (a ∈ A) instead of (9.41), we assume ρ ∈ S[Z]. A further generalization of the problem is obtained by introducing a cost function for the flow boundary ∂ξ rather than merely imposing the constraint ∂ξ ∈ B. Namely, with a function f : RV → R ∪ {+∞} we add a new term f (∂ξ) to the objective function, thereby imposing the constraint ∂ξ ∈ B = dom f implicitly. The aforementioned nice properties are maintained if f is a polyhedral M-convex function. Such a problem described by some f ∈ M[R → R] is called the M-convex submodular flow problem. M-convex submodular flow problem MSFP2 (linear arc cost)  Minimize Γ2 (ξ) = γ(a)ξ(a) + f (∂ξ) (9.42) a∈A

subject to c(a) ≤ ξ(a) ≤ c(a)

(a ∈ A),

∂ξ ∈ dom f, ξ(a) ∈ R (a ∈ A).

(9.43) (9.44) (9.45)

Note that the M-convex submodular flow problem with a {0, +∞}-valued f reduces to the submodular flow problem MSFP1 . In the integer-flow version of the problem we assume c : A → Z ∪ {+∞}, c : A → Z ∪ {−∞}, and f ∈ M[Z → R] (or f ∈ M[Z|R → R]). A still further generalization is possible by replacing the linear arc cost in Γ2 with a separable convex function. Namely, using univariate polyhedral convex functions fa ∈ C[R → R] (a ∈ A), we consider a∈A fa (ξ(a)) instead of a∈A γ(a)ξ(a) to obtain MSFP3 below, a special case of MCFP3 with f being M-convex. M-convex submodular flow problem MSFP3 (nonlinear arc cost)  fa (ξ(a)) + f (∂ξ) (9.46) Minimize Γ3 (ξ) = a∈A

subject to ξ(a) ∈ dom fa

(a ∈ A),

∂ξ ∈ dom f, ξ(a) ∈ R (a ∈ A).

(9.47) (9.48) (9.49)

In the integer-flow version of the problem we assume f ∈ M[Z → R] and fa ∈ C[Z → R] for a ∈ A (or f ∈ M[Z|R → R] and fa ∈ C[Z|R → R] for a ∈ A). Obviously, MSFP2 is a special case of MSFP3 with  γ(a)t (t ∈ [c(a), c(a)]), fa (t) = (9.50) +∞ (otherwise). The converse is also true; i.e., MSFP3 can be put into a problem of the form of MSFP2 , as is explained in Note 9.8. Throughout this chapter we assume ρ(V ) = 0,

dom f ⊆ {x ∈ RV | x(V ) = 0},

(9.51)

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since ∂ξ(V ) = 0 for any flow ξ and ∂ξ ∈ dom f = B = B(ρ) is imposed. In subsequent sections we will see that the optimality criteria in terms of potentials and negative cycles, as well as efficient algorithms for the conventional minimum cost flow problem MCFP0 , can be generalized for the M-convex submodular flow problem. Note 9.8. The problem MSFP3 on G = (V, A) can be written in the form of ˜ = (V˜ , A). ˜ We replace each arc a = (u, v) ∈ A with a MSFP2 on a larger graph G + − − pair of arcs, a = (u, va ) and a = (va+ , v), where va+ and va− are newly introduced vertices. Accordingly, we have A˜ = {a+ , a− | a ∈ A} and V˜ = V ∪ {va+ , va− | a ∈ A}. For each a ∈ A we consider a function f˜a : R2 → R ∪ {+∞} given by  fa (t) (t + s = 0), f˜a (t, s) = +∞ (otherwise) ˜ and define f˜ : RV → R ∪ {+∞} by  x(va+ ), x ˜(va− )) + f (˜ x|V ) f˜a (˜ f˜(˜ x) =

˜

(˜ x ∈ RV ),

a∈A

where x ˜|V denotes the restriction of x ˜ to V . For a flow ξ˜ : A˜ → R, we have ˜ − ) if (∂ ξ(v ˜ + ), ∂ ξ(v ˜ − )) ∈ dom f˜a . The problem MSFP3 is thus reduced ˜ + ) = ξ(a ξ(a a a ˜ = f˜(∂ ξ). ˜ Note that, if f ∈ M[R → R] ˜ 2 (ξ) to MSFP2 with the objective function Γ and fa ∈ C[R → R] for a ∈ A, then f˜ ∈ M[R → R]. Note 9.9. The cost function Γ3 of MSFP3 consists of two terms, the separable arc cost a∈A fa (ξ(a)) and the M-convex boundary cost f (∂ξ). Noting that the former is M -convex, one might be tempted to consider a (nonseparable) M -convex cost function defined on the arc set. The integer-flow version of such a problem, however, contains the Hamiltonian path problem, a well-known NP-complete problem, as a special case. Suppose that we want to check for the existence of an (s, t)-Hamiltonian path in a directed graph G = (V, A), where we may assume s = t ∈ V and δ − s = δ + t = ∅. ˜ = (V˜ , A) ˜ by replacing each arc a = (u, v) ∈ We construct another directed graph G A with three arcs connected in series: a+ = (u, va− ),

a0 = (va− , va+ ),

a− = (va+ , v),

where va+ and va− are newly introduced vertices. Hence, V˜ = V ∪ {va+ , va− | a ∈ A} and A˜ = A+ ∪ A0 ∪ A− , with A+ = {a+ | a ∈ A}, A0 = {a0 | a ∈ A}, and A− = {a− | a ∈ A}. We consider three matroids, say, M+ , M− , and M0 on A+ , A− , and A0 , respectively. M+ is a partition matroid in which B + ⊆ A+ is a base if and only if |B + ∩ {a+ | ∂ + a = v}| = 1

(∀ v ∈ V ),

M− is another partition matroid defined similarly (with + replaced with −), and M0 is the graphic matroid in which B 0 ⊆ A0 is a base if and only if {a ∈ A | a0 ∈ B 0 } is

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a tree of the original graph G. Let Q be the set of characteristic vectors of a subset ˜ ∩ A− is a base of M− , and B ˜ ∩ A0 is ˜ of A˜ such that B ˜ ∩ A+ is a base of M+ , B B 0 ˜ ˜ ˜ ˜ an independent set of M . Then a {0, 1}-flow ξ in G with ξ ∈ Q and ∂ ξ = χs − χt corresponds to an (s, t)-Hamiltonian path in G. Since Q is an M -convex set, the constraint ξ˜ ∈ Q can be represented by a {0, +∞}-valued M -convex cost function ˜ on the arc set A.

9.3

Feasibility of Submodular Flow Problem

The feasibility of the submodular flow problem MSFP1 is investigated here. Recall that we are given a graph G = (V, A), an upper capacity c : A → R∪{+∞}, a lower capacity c : A → R ∪ {−∞}, and a submodular set function ρ : 2V → R ∪ {+∞}, where c(a) ≥ c(a) for a ∈ A and ρ(∅) = ρ(V ) = 0. A feasible flow means a function ξ : A → R that satisfies c(a) ≤ ξ(a) ≤ c(a) ∂ξ ∈ B(ρ).

(a ∈ A),

(9.52) (9.53)

The problem MSFP1 is said to be feasible if it admits a feasible flow. In section 9.1.2 we considered (9.52) to obtain Theorem 9.3. We now combine (9.52) and (9.53) for the feasibility of MSFP1 . Theorem 9.10 (Feasibility). A submodular flow problem MSFP1 is feasible if and only if c(Δ− X) − c(Δ+ X) + ρ(X) ≥ 0 (∀ X ⊆ V ). (9.54) Moreover, if c, c, and ρ are integer valued and the problem is feasible, there exists an integer-valued feasible flow ξ : A → Z. Proof. Let κ be the cut capacity function defined by (9.16). By Theorem 9.3 a feasible flow exists if and only if B(κ) ∩ B(ρ) = ∅. The latter condition is equivalent to κ(V \ X) + ρ(X) ≥ 0 (∀ X ⊆ V ) by Edmonds’s intersection theorem (Theorem 4.18) and further to (9.54) by κ(V \ X) = c(Δ− X) − c(Δ+ X). In a feasible problem with integer-valued c, c, and ρ, both B(κ) and B(ρ) are integral base polyhedra (integral M-convex polyhedra), and B(κ) ∩ B(ρ) ∩ ZV is nonempty by (4.32). Then (9.19) in Theorem 9.3 guarantees the existence of an integer flow ξ : A → Z with ∂ξ ∈ B(κ) ∩ B(ρ) ∩ ZV . Note 9.11. The necessity of (9.54) is easy to see. For any X ⊆ V , the net amount of flow entering X is equal to zero:   ξ(a) − ξ(a) + ∂ξ(X) = 0, (9.55) a∈Δ− X

a∈Δ+ X

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and the constraints (9.52) and (9.53) should be satisfied: ξ(a) ≤ c(a) (a ∈ Δ− X),

ξ(a) ≥ c(a) (a ∈ Δ+ X),

∂ξ(X) ≤ ρ(X).

(9.56)

Combining these two yields (9.54). Theorem 9.10 claims that this “obvious” necessary condition is in fact sufficient. Note 9.12. In a feasible submodular flow problem, the set of boundaries of feasible flows, ∂Ξ = {∂ξ | ξ : feasible flow}, is an M2 -convex polyhedron, and it is an integral M2 -convex polyhedron if c, c, and ρ are integer valued. This can be seen from the proof of Theorem 9.10. The maximum submodular flow problem is to find a feasible flow ξ that maximizes ξ(a0 ) for a specified arc a0 ∈ A. Maximum submodular flow problem maxSFP Maximize ξ(a0 )

(9.57)

subject to c(a) ≤ ξ(a) ≤ c(a) ∂ξ ∈ B(ρ), ξ(a) ∈ R

(a ∈ A),

(a ∈ A).

(9.58) (9.59) (9.60)

A max-flow min-cut theorem holds for this problem. Note that for any X ⊂ V with a0 ∈ Δ+ X we have an “obvious” inequality:   ξ(a0 ) = ξ(a) − ξ(a) + ∂ξ(X) a∈Δ− X

a∈Δ+ X\{a0 }

≤ c(Δ− X) − c(Δ+ X \ {a0 }) + ρ(X), by (9.55) and (9.56). Theorem 9.13 (Max-flow min-cut theorem). For a feasible maximum submodular flow problem maxSFP, max{ξ(a0 ) | (9.58), (9.59), (9.60)} ξ @ ? = min c(a0 ), min{c(Δ− X) − c(Δ+ X \ {a0 }) + ρ(X) | a0 ∈ Δ+ X} , (9.61) X

where this common value can be +∞. If c, c, and ρ are integer valued and (9.61) is finite, there exists an integer-valued maximum flow ξ : A → Z. Proof. Divide the arc a0 = (u, v) into two arcs in series, say, a0 = (u, w) and ˜ = (V˜ , A) ˜ the resulting graph, where V˜ = V ∪ {w} a0 = (w, v), and denote by G and A˜ = A ∪ {a0 }. Define the capacities of a0 by c(a0 ) = t and c(a0 ) = +∞ with a parameter t, and let ρ be defined for all subsets of V˜ by ρ(X ∪ {w}) = ρ(X) for X ⊆ V . The maximum in (9.61) is equal to the maximum (or supremum) of t such

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˜ = (V˜ , A) ˜ is feasible. With this relationship, that the submodular flow problem on G Theorem 9.10 implies (9.61) as well as the integrality assertion. If ξ and X attain the maximum and the minimum in (9.61), respectively, and if ξ(a0 ) < c(a0 ), then we have ∂ξ(X) = ρ(X),

9.4

ξ(a) = c(a) (a ∈ Δ− X),

ξ(a) = c(a) (a ∈ Δ+ X \ {a0 }). (9.62)

Optimality Criterion by Potentials

In section 9.1.3 we saw a potential criterion for optimality (Theorem 9.4) for the minimum cost flow problem MCFP3 . Since the M-convex submodular flow problem MSFP3 is a special case of MCFP3 , the following optimality criterion for MSFP3 is immediate from Theorem 9.4. Theorem 9.14 (Potential criterion). In the M-convex submodular flow problem MSFP3 with fa ∈ C[R → R] (a ∈ A) and f ∈ M[R → R], we have the following. (1) For a feasible flow ξ : A → R, the two conditions (OPT) and (POT) below are equivalent. (OPT) ξ is an optimal flow. (POT) There exists a potential p : V → R such that (i) ξ(a) ∈ arg min fa [δp(a)] for every a ∈ A, and (ii) ∂ξ ∈ arg min f [−p]. (2) Suppose that a potential p : V → R satisfies (i) and (ii) above for an optimal flow ξ. A feasible flow ξ  is optimal if and only if (i) ξ  (a) ∈ arg min fa [δp(a)] for every a ∈ A, and (ii) ∂ξ  ∈ arg min f [−p]. A comment is in order on the role of the M-convexity of f . Since fa is a univariate convex function for every a ∈ A, condition (i) in (POT) can be expressed in terms of directional derivatives as: ξ(a) ∈ arg min fa [δp(a)] ⇐⇒ fa (ξ(a); d) + d[p(∂ + a) − p(∂ − a)] ≥ 0

(d = ±1).

(9.63)

If f is M-convex, condition (ii) in (POT) can also be expressed in terms of directional derivatives as: ∂ξ ∈ arg min f [−p] ⇐⇒ f  (∂ξ; −χu + χv ) + p(u) − p(v) ≥ 0

(∀ u, v ∈ V ),

(9.64)

by the M-optimality criterion in Theorem 6.52 (1). These expressions show how the conditions in (POT) can be verified efficiently for a given p. It is also mentioned that these expressions lead to another optimality criterion in terms of negative cycles, to be established in section 9.5, and furthermore to the cycle-canceling algorithm for the M-convex submodular flow problem, to be explained in section 10.4.3.

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261

An alternative representation of condition (ii) in (POT) is obtained from the M-convexity of f . The function conjugate to f , say, g, is a polyhedral L-convex function with g(p + 1) = g(p) for all p (by Theorem 8.4 and (9.51)). It follows from (3.30) and the L-optimality criterion (Theorem 7.33 (1)) that x ∈ arg min f [−p] ⇐⇒ p ∈ arg min g[−x]   g (p; χX ) − x(X) ≥ 0 (∀ X ⊆ V ), ⇐⇒ g  (p; 1) − x(V ) = 0. This shows that arg min f [−p] coincides with the base polyhedron B(gp ) associated with the set function gp defined by gp (X) = g  (p; χX )

(X ⊆ V ),

which is submodular by Theorem 7.43 (1). Hence, ∂ξ ∈ arg min f [−p] ⇐⇒ ∂ξ ∈ B(gp ).

(9.65)

This expression is used in the primal-dual algorithm for the M-convex submodular flow problem, to be explained in section 10.4.4. We go on to discuss integrality properties of the M-convex submodular flow problem. This generalizes the well-known facts (Theorem 9.6) for the minimum cost flow problem MCFP0 . Recall the notation M[Z|R → R] and M[R → R|Z] for the sets of integral and dual-integral polyhedral M-convex functions, respectively. Theorem 9.15. Suppose that an optimal solution exists in the M-convex submodular flow problem MSFP3 with fa ∈ C[R → R] (a ∈ A) and f ∈ M[R → R]. (1) The set of the boundaries of optimal flows, ∂Ξ∗ = {∂ξ | ξ : optimal flow}, is an M2 -convex polyhedron, and the set of optimal potentials, Π∗ = {p | p : optimal potential }, is an L-convex polyhedron. (2) [Primal integrality] If fa ∈ C[Z|R → R] (a ∈ A) and f ∈ M[Z|R → R], then ∂Ξ∗ is an integral M2 -convex polyhedron, and there exists an integer-valued optimal flow ξ : A → Z. (3) [Dual integrality] If fa ∈ C[R → R|Z] (a ∈ A) and f ∈ M[R → R|Z], then Π∗ is an integral L-convex polyhedron, and there exists an integer-valued optimal potential p : V → Z. Proof. (1) Let p be an optimal potential. Since arg min fa [δp(a)] forms an interval, say, [c∗ (a), c∗ (a)]R , condition (i) in (POT) of Theorem 9.14 can be expressed as c∗ (a) ≤ ξ(a) ≤ c∗ (a) (a ∈ A). Just as in (9.16) and (9.18), the set of ∂ξ for such ξ coincides with the base polyhedron B(κ∗ ) for κ∗ defined by κ∗ (X) = c∗ (Δ+ X) − c∗ (Δ− X)

(X ⊆ V ).

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Combining this with (9.65) we obtain ∂Ξ∗ = B(κ∗ ) ∩ B(gp ), which is an M2 -convex polyhedron. Now let ξ be an optimal flow. Potentials p satisfying (i) in (POT) form an L-convex polyhedron, say, D1 , by (9.63), whereas those satisfying (ii) in (POT) form another L-convex polyhedron D2 by (9.64). Therefore, Π∗ = D1 ∩ D2 is an L-convex polyhedron. (2) Both B(κ∗ ) and B(gp ) are integral M-convex polyhedra. The integrality ∗ of ∂Ξ = B(κ∗ ) ∩ B(gp ) follows from (4.32). (3) Both D1 and D2 are integral L-convex polyhedra by Theorem 6.61 (1). The integrality of Π∗ = D1 ∩ D2 follows from Theorem 5.7. For linear arc cost, with fa given by (9.50), the integrality conditions are simplified as follows: fa ∈ C[Z|R → R] ⇐⇒ c(a), c(a) ∈ Z,

(9.66)

fa ∈ C[R → R|Z] ⇐⇒ γ(a) ∈ Z.

(9.67)

Finally, we state the optimality criterion for the integer-flow version of the M-convex submodular flow problem MSFP3 . This is a corollary of Theorems 9.14 and 9.15. Theorem 9.16 (Potential criterion). Consider the M-convex submodular integerflow problem MSFP3 with fa ∈ C[Z → R] (a ∈ A) and f ∈ M[Z → R]. (1) For a feasible integer flow ξ : A → Z, the two conditions (OPT) and (POT) below are equivalent. (OPT) ξ is an optimal integer flow. (POT) There exists a potential p : V → R such that (i) ξ(a) ∈ arg min fa [δp(a)] for every a ∈ A, and (ii) ∂ξ ∈ arg min f [−p]. (2) Suppose that a potential p : V → R satisfies (i) and (ii) above for an optimal integer flow ξ. A feasible integer flow ξ  is optimal if and only if (i) ξ  (a) ∈ arg min fa [δp(a)] for every a ∈ A, and (ii) ∂ξ  ∈ arg min f [−p]. (3) The set of the boundaries of optimal integer flows, ∂Ξ∗ = {∂ξ | ξ : optimal integer flow}, is an M2 -convex set. (4) If the cost functions are integer valued, i.e., if fa ∈ C[Z → Z] (a ∈ A) and f ∈ M[Z → Z], then there exists an integer-valued potential p : V → Z in (POT). Moreover, the set of integer-valued optimal potentials, Π∗ = {p | p : integer-valued optimal potential}, is an L-convex set. In connection to (i) and (ii) in (POT) in Theorem 9.16, note the equivalences

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263

ξ(a) ∈ arg min fa [δp(a)] ⇐⇒ fa (ξ(a) + d) − fa (ξ(a)) + d[p(∂ + a) − p(∂ − a)] ≥ 0 ∂ξ ∈ arg min f [−p] ⇐⇒ Δf (∂ξ; v, u) + p(u) − p(v) ≥ 0

(d = ±1),

(∀ u, v ∈ V ).

(9.68) (9.69)

These are the discrete counterparts of (9.63) and (9.64). Note 9.17. The Fenchel-type duality theorem for M-convex functions (Theorem 8.21) is essentially equivalent to the optimality criterion for the M-convex submodular integer-flow problem (Theorem 9.16). See section 9.1.4 and note that, for an M-convex function f1 and an M-concave function h2 , f (x1 , x2 ) = f1 (x1 ) − h2 (−x2 ) is an M-convex function.

9.5

Optimality Criterion by Negative Cycles

The optimality of an M-convex submodular flow can also be characterized by the nonexistence of negative cycles in an auxiliary network. This fact leads to the cycle-canceling algorithm to be described in section 10.4.3.

9.5.1

Negative-Cycle Criterion

We consider the M-convex submodular flow problem MSFP2 with M-convex boundary cost and linear arc cost. This is not restrictive, since MSFP3 , having nonlinear convex arc cost, can be put in the form of MSFP2 , as explained in Note 9.8. We consider real-valued flows and then integer-valued flows. We assume f ∈ M[R → R] in considering real-valued flows. For a feasible flow ξ : A → R, we define an auxiliary network as follows. Let Gξ = (V, Aξ ) be a directed graph with vertex set V and arc set Aξ = A∗ξ ∪ Bξ∗ ∪ Cξ consisting of three disjoint parts: A∗ξ = {a | a ∈ A, ξ(a) < c(a)}, Bξ∗ = {a | a ∈ A, c(a) < ξ(a)}

(a: reorientation of a),

Cξ = {(u, v) | u, v ∈ V, u = v, ∃ α > 0 : ∂ξ − α(χu − χv ) ∈ domR f }. (9.70) We define a function ξ : Aξ → R, representing arc lengths, by ⎧ (a ∈ A∗ξ ), ⎨ γ(a) (a ∈ Bξ∗ , a ∈ A), −γ(a) ξ (a) = ⎩  f (∂ξ; −χu + χv ) (a = (u, v) ∈ Cξ ).

(9.71)

We refer to (Gξ , ξ ) as the auxiliary network . We call a directed cycle of negative length a negative cycle. The following theorem gives an optimality criterion in terms of negative cycles. Theorem 9.18 (Negative-cycle criterion). For a feasible flow ξ : A → R to the M-convex submodular flow problem MSFP2 with f ∈ M[R → R], the conditions (OPT) and (NNC) below are equivalent.

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Chapter 9. Network Flows (OPT) ξ is an optimal flow. (NNC) There exists no negative cycle in (Gξ , ξ ) with ξ of (9.71).

Proof. As is well known in network flow theory, (NNC) is equivalent to the existence of a potential p : V → R such that ξ (a) + p(∂ + a) − p(∂ − a) ≥ 0

(a ∈ Aξ ).

By (9.31), (9.32), (9.64), and the definition (9.71) of ξ , this condition is equivalent to conditions (i) and (ii) of (POT) in Theorem 9.14. Hence follows the equivalence of (NNC) and (OPT) by Theorem 9.14.

Note 9.19. In a problem with dual integrality the arc length ξ is integer valued. The integrality of ξ (a) for a ∈ A∗ξ ∪ Bξ∗ is due to (9.67) and that for a ∈ Cξ is by Theorem 6.61 (1). For integer-valued ξ , we can take an integer-valued p in the proof of Theorem 9.18. Next we consider the integer-flow problem under the assumptions c : A → Z ∪ {+∞},

c : A → Z ∪ {−∞},

f ∈ M[Z → R].

(9.72)

For a feasible integer flow ξ : A → Z, we define an auxiliary network (Gξ , ξ ) in a similar manner, while modifying the definitions of Cξ and ξ to Cξ = {(u, v) | u, v ∈ V, u = v, ∂ξ − (χu − χv ) ∈ domZ f }, ⎧ (a ∈ A∗ξ ), ⎨ γ(a) −γ(a) (a ∈ Bξ∗ , a ∈ A), ξ (a) = ⎩ Δf (∂ξ; v, u) (a = (u, v) ∈ Cξ ).

(9.73) (9.74)

Theorem 9.20 (Negative-cycle criterion). For a feasible integer flow ξ : A → Z to the M-convex submodular integer flow problem MSFP2 with (9.72), the conditions (OPT) and (NNC) below are equivalent. (OPT) ξ is an optimal flow. (NNC) There exists no negative cycle in (Gξ , ξ ) with ξ of (9.74). Proof. This is similar to the proof of Theorem 9.18. Note, however, that Theorem 9.16 is used here in place of Theorem 9.14.

Note 9.21. The M-convex intersection problem introduced in section 8.2.1 can be formulated as an M-convex submodular flow problem. Given two M-convex functions f1 , f2 : ZV → R ∪ {+∞}, we consider an M-convex submodular flow problem on the bipartite graph G = (V1 ∪ V2 , A) in Fig. 9.3, where V1 and V2 are copies of V and A = {(v1 , v2 ) | v ∈ V } with v1 ∈ V1 and v2 ∈ V2 denoting the copies of v ∈ V . The boundary cost function f : ZV1 × ZV2 → R ∪ {+∞} is defined by f (x1 , x2 ) = f1 (x1 ) + f2 (−x2 ) for x1 ∈ ZV1 and x2 ∈ ZV2 , whereas the arc costs are

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V1 f1 (x1 )

265

V2 f2 (−x2 )

Figure 9.3. Submodular flow problem for M-convex intersection problem.

identically zero without capacity constraints. Since x1 = −x2 if (x1 , x2 ) = ∂ξ for a flow ξ in this network, the M-convex submodular flow problem is equivalent to minimizing f1 (x) + f2 (x). The negative-cycle optimality criterion (Theorem 9.20) for this M-convex submodular flow problem yields the M2 -optimality criterion in Theorem 8.33. This argument shows also that the M2 -optimality criterion can be verified in polynomial time.

9.5.2

Cycle Cancellation

The negative-cycle optimality criterion states that the existence of a negative cycle implies the nonoptimality of a feasible flow. This suggests the possibility of improving a nonoptimal feasible flow by the cancellation of a suitably chosen negative cycle. Let us consider the integer-flow problem with (9.72). Suppose that negative cycles exist in the auxiliary network (Gξ , ξ ) for a feasible integer flow ξ, where the arc length ξ is defined by (9.74). Choose a negative cycle with the smallest number of arcs and let Q (⊆ Aξ ) be the set of its arcs. Modifying the flow ξ along Q we obtain a new integer flow ξ defined by ⎧ ⎨ ξ(a) + 1 (a ∈ Q ∩ A∗ξ ), ξ(a) − 1 (a ∈ Q ∩ Bξ∗ ), ξ(a) = (9.75) ⎩ ξ(a) (otherwise). The following theorem shows that ξ is a feasible flow with an improvement in the objective function  Γ2 (ξ) = γ(a)ξ(a) + f (∂ξ). a∈A

This gives an alternative proof for “(OPT) ⇒ (NNC),” which has already been established in Theorem 9.20 with the aid of Theorem 9.16. Theorem 9.22. For a feasible integer flow ξ to the M-convex submodular integerflow problem MSFP2 with (9.72), let Q be a negative cycle with the smallest number of arcs in (Gξ , ξ ). Then ξ in (9.75) is a feasible integer flow and Γ2 (ξ) ≤ Γ2 (ξ) + ξ (Q) < Γ2 (ξ).

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The rest of this section is devoted to the proof of Theorem 9.22. The key ingredient of the proof is the unique-min condition, defined as follows. For a pair (x, y) of integer vectors satisfying x ∈ domZ f and ||x− y||∞ = 1, we consider a bipartite graph G(x, y) = (V + , V − ; E) with vertex sets V + = supp+ (x − y) and V − = supp− (x − y) and arc set E = {(u, v) | u ∈ V + , v ∈ V − , x − χu + χv ∈ domZ f } and associate c(u, v) = Δf (x; v, u) with arc (u, v) ∈ E as its weight. We say that (x, y) satisfies the unique-min condition if there exists in G(x, y) exactly one minimum-weight perfect matching with respect to c. Denote by fˇ(x, y) the minimum weight of a perfect matching in G(x, y), where ˇ f (x, y) = +∞ if no perfect matching exists. Proposition 6.25 shows f (y) − f (x) ≥ fˇ(x, y) for any x ∈ domZ f and y ∈ ZV . The unique-min condition is a sufficient condition for this inequality to be an equality. Proposition 9.23. Let f ∈ M[Z → R] be an M-convex function, and assume x ∈ domZ f , y ∈ ZV , and ||x−y||∞ = 1. If (x, y) satisfies the unique-min condition, then y ∈ domZ f and f (y) − f (x) = fˇ(x, y). (9.76) Proof. The set function ω defined by ω(X) = −f (x∧y +χX ) (X ⊆ V ) is a valuated matroid; see (2.77). The present claim is a reformulation of the unique-max lemma for valuated matroids (see Theorem 5.2.35 in Murota [146]). The following proposition gives a necessary and sufficient condition for a bipartite graph to have a unique minimum-weight perfect matching. It also shows that the unique-min condition for a pair of integer vectors can be checked by an efficient algorithm. Proposition 9.24. Let G = (V + , V − ; E) be a bipartite graph with |V + | = |V − | (= m) and c : V + ×V − → R∪{+∞} be a weight function such that c(u, v) < +∞ ⇐⇒ (u, v) ∈ E. There exists a unique minimum-weight perfect matching if and only if there exist a potential pˆ : V + ∪V − → R and orderings of vertices V + = {u1 , . . . , um } and V − = {v1 , . . . , vm } such that ⎧ ⎨ = 0 (1 ≤ i = j ≤ m), ≥ 0 (1 ≤ j < i ≤ m), c(ui , vj ) + pˆ(ui ) − pˆ(vj ) ⎩ > 0 (1 ≤ i < j ≤ m).

(9.77)

Proof. This follows from the complementarity (Theorem 3.10 (3)) in the linear program formulation in the proof of Proposition 3.14. The following is the key fact. Note that ∂ξ ∈ domZ f and ||∂ξ − ∂ξ||∞ = 1.

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267

Proposition 9.25. (∂ξ, ∂ξ) satisfies the unique-min condition. Proof. Consider the bipartite graph G(∂ξ, ∂ξ) = (V + , V − ; E), where V + = supp+ (∂ξ − ∂ξ), V − = supp− (∂ξ − ∂ξ), and E = {(u, v) | u ∈ V + , v ∈ V − , ∂ξ − χu + χv ∈ domZ f }. We have |V + | = |V − | = m for m = ||∂ξ − ∂ξ||1 /2 and the weight of arc (u, v) equal to Δf (∂ξ; v, u). We may think of G(∂ξ, ∂ξ) as a subgraph of the graph Gξ of section 9.5.1 by regarding E as a subset of Cξ in (9.73). Then Q ∩ Cξ determines a perfect matching in G(∂ξ, ∂ξ). Let M = {(ui , vi ) | i = 1, . . . , m} be a minimum-weight perfect matching in G(∂ξ, ∂ξ) and pˆ be an optimal potential in Proposition 3.14. Note that M is a subset of Cξ∗ = {(u, v) | u ∈ V + , v ∈ V − , Δf (∂ξ; v, u) + pˆ(u) − pˆ(v) = 0}. Regarding M as a subset of Cξ , we define Q = (Q\Cξ )∪M . Since M is a minimumweight perfect matching, Q ∩ Cξ is a perfect matching, and ξ (a) = Δf (∂ξ; v, u) for a = (u, v) ∈ Cξ , we have ξ (M ) ≤ ξ (Q ∩ Cξ ), from which follows ξ (Q ) = ξ (Q) + [ξ (M ) − ξ (Q ∩ Cξ )] ≤ ξ (Q) < 0.

(9.78)

Since Q is a union of disjoint cycles with |Q | = |Q| and Q is a negative cycle with the smallest number of arcs, (9.78) implies that Q is also a negative cycle with the smallest number of arcs. To prove by contradiction, suppose that (∂ξ, ∂ξ) does not satisfy the uniquemin condition. Since (ui , vi ) ∈ Cξ∗ for i = 1, . . . , m, it follows from Proposition 9.24 that there exist distinct indices ik (k = 1, . . . , q; q ≥ 2) such that (uik , vik+1 ) ∈ Cξ∗ for k = 1, . . . , q, where iq+1 = i1 . That is, p(uik ) + pˆ(vik+1 ) Δf (∂ξ; vik+1 , uik ) = −ˆ

(k = 1, . . . , q).

On the other hand, we have Δf (∂ξ; vik , uik ) = −ˆ p(uik ) + pˆ(vik )

(k = 1, . . . , q).

It then follows that q 

Δf (∂ξ; vik+1 , uik ) =

k=1

q 

Δf (∂ξ; vik , uik );

k=1

i.e., q  k=1

ξ (uik , vik+1 ) =

q  k=1

ξ (uik , vik ).

(9.79)

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For k = 1, . . . , q, let P  (vik+1 , uik ) denote the path on Q from vik+1 to uik , and let Qk be the directed cycle consisting of arc (uik , vik+1 ) and path P  (vik+1 , uik ). Obviously, 8 q 9 q : :   Qk = P (vik+1 , uik ) ∪ {(uik , vik+1 ) | k = 1, . . . , q}, k=1

k=1

where the union here (and also below) means the multiset union, counting the number of occurrences of elements. A simple but crucial observation is that 8 q 9 :  P (vik+1 , uik ) ∪ {(uik , vik ) | k = 1, . . . , q} = q  · Q (9.80) k=1

for some integer q  with 1 ≤ q  < q. Hence, q 

ξ (Qk ) =

k=1

=

q  k=1 q 

ξ (P  (vik+1 , uik )) + ξ (P  (vik+1 , uik )) +

k=1 

q  k=1 q 

ξ (uik , vik+1 ) ξ (uik , vik )

k=1

= q · ξ (Q ) < 0, where (9.79) and (9.78) are used. This implies that ξ (Qk ) < 0 for some k, which, however, is a contradiction, since Qk has a smaller number of arcs than Q . This completes the proof. Proof of Theorem 9.22: It follows from Propositions 9.25 and 9.23 as well as the definition of fˇ that f (∂ξ) = f (∂ξ) + fˇ(∂ξ, ∂ξ) ≤ f (∂ξ) + ξ (Q ∩ Cξ ), whereas

 a∈A

γ(a)ξ(a) =



γ(a)ξ(a) + ξ (Q ∩ (A∗ξ ∪ Bξ∗ )).

a∈A

Adding these two results in Γ2 (ξ) ≤ Γ2 (ξ) + ξ (Q).

9.6

Network Duality

Transformation by a network is one of the most important operations for M-convex and L-convex functions. A given pair of M-convex and L-convex functions defined on entrance vertices of a network is transformed through the network to another pair of M-convex and L-convex functions on exit vertices. Moreover, if the functions in the given pair are conjugate to each other, the resulting pair is also conjugate. This fact reveals a deeper intrinsic relationship of M-/L-convexity to network flow, partly discussed in section 2.2. The theorems as well as their implications are stated in section 9.6.1, and the proofs are given in section 9.6.2.

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269 (fa , ga ) -

S

j *

f (x) g(p)

-

1

j 

s q

j

N

T

1

O

W

f˜(y)



g˜(q)

> -

(ξ(a), η(a))

Figure 9.4. Transformation by a network .

9.6.1

Transformation by Networks

We first deal with functions of the Z → Z type, integer-valued functions defined on integer points, and then functions of other types, Z → R and R → R. Let G = (V, A; S, T ) be a directed graph with vertex set V , arc set A, entrance set S, and exit set T , where S and T are disjoint subsets of V ; see Fig. 9.4 for an illustration. For each a ∈ A, the costs of integer-valued flow and tension are represented, respectively, by functions fa : Z → Z ∪ {+∞} and ga : Z → Z ∪ {+∞}. Given functions f, g : ZS → Z ∪ {+∞} associated with the entrance set S of the network, we define functions f˜, g˜ : ZT → Z ∪ {±∞} on the exit set T by ) ( )  ) f˜(y) = inf f (x) + fa (ξ(a))) ∂ξ = (x, −y, 0), ) ξ,x a∈A * ξ ∈ ZA , (x, −y, 0) ∈ ZS × ZT × ZV \(S∪T ) ( g˜(q) = inf

η,p,r

g(p) +

 a∈A

) ) ) ga (η(a))) η = −δ(p, q, r), ) V \(S∪T )

η ∈ Z , (p, q, r) ∈ Z × Z × Z A

S

T

(y ∈ ZT ),

(9.81)

* (q ∈ ZT ).

(9.82)

We may think of f˜(y) as the minimum cost to meet a demand specification y at the exit, where the cost consists of two parts, the cost f (x) of supply or production of x at the entrance and the cost a∈A fa (ξ(a)) of transportation through arcs; the sum of these is to be minimized over varying supply x and flow ξ subject to the flow conservation constraint ∂ξ = (x, −y, 0). A similar interpretation is possible for g˜(q). We regard f˜ and g˜ as the results of transformations of f and g by the network ; (9.81) and (9.82) are called transformations of flow type and of potential type, respectively.

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Chapter 9. Network Flows

The following theorem reveals the harmonious relationship between network flow and M-/L-convexity by which a conjugate pair of M-convex and L-convex functions is transformed to another conjugate pair of M-convex and L-convex functions. Note that C[Z → Z] denotes the set of univariate integer-valued discrete convex functions, and • means the discrete Legendre–Fenchel transformation (8.11)Z . Theorem 9.26. Assume fa , ga ∈ C[Z → Z] for each a ∈ A. For f, g : ZS → Z ∪ {+∞}, let f˜, g˜ : ZT → Z ∪ {±∞} be the functions induced by G = (V, A; S, T ) according to (9.81) and (9.82), where it is assumed that f˜ > −∞, f˜ ≡ +∞, g˜ > −∞, and g˜ ≡ +∞. (1) f f (2) g g (3) f

∈ M[Z → Z] =⇒ f˜ ∈ M[Z → Z], and ∈ M [Z → Z] =⇒ f˜ ∈ M [Z → Z]. ∈ L[Z → Z] =⇒ g˜ ∈ L[Z → Z], and ∈ L [Z → Z] =⇒ g˜ ∈ L [Z → Z]. ∈ M [Z → Z], g ∈ L [Z → Z], g = f • , ga = fa • (a ∈ A) =⇒ g˜ = f˜• .

We explain the implications of this theorem by considering three special cases. The first special case is a well-known construction in matroid theory, induction of a matroid through a graph. Given a graph G = (V, A; S, T ) and a matroid (S, B) on S with base family B, let B˜ be the family of subsets of T that can be linked with some base of (S, B) by a vertex-disjoint linking in G. Then B˜ forms the base family of a matroid on T , which is referred to as the matroid induced from (S, B) through G. To formulate this as a special case of Theorem 9.26 (1), we split each vertex ¯ = (V  ∪ V  , A; ¯ S  , T  ) with v ∈ V into two copies, v  and v  , to consider a graph G A¯ = {(u , v  ) | (u, v) ∈ A} ∪ {(v  , v  ) | v ∈ V }, 

where S  = {v  | v ∈ S} and T  = {v  | v ∈ T }. Let f : ZS → Z ∪ {+∞} be the indicator function of the set of the characteristic vectors of bases, i.e., f = δB for ¯ define fa to be the indicator function B = {χX  | X ∈ B}, and, for each arc a ∈ A, T  ˜ of {0, 1}. Then the induced function f : Z → Z ∪ {+∞} represents the family B˜ ˜ = {χY  | Y ∈ B}. ˜ Then the M-convexity of f˜ stated in the sense that f˜ = δB˜ for B ˜ in Theorem 9.26 (1) shows that (T, B) is a matroid. The second case is where (S, T ) = (V, ∅). Then the induced functions f˜ and g˜ are constants, having no arguments, and the conjugacy asserted in Theorem 9.26 (3) amounts to a min-max relation inf

ξ∈ZA ,x∈ZV

{Φ(ξ, x) | ∂ξ = x} =

sup η∈ZA ,p∈ZV

{Ψ(η, p) | η = −δp}

for Φ(ξ, x) = f (x) +



fa (ξ(a))

(ξ ∈ ZA , x ∈ ZV ),

a∈A

Ψ(η, p) = −g(p) −



a∈A

ga (η(a))

(η ∈ ZA , p ∈ ZV ).

(9.83)

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271

This is the discrete counterpart of (9.36), showing the duality nature of the assertion of Theorem 9.26 (3). The third case is where S = ∅. Then the induced functions are ) ( * )  ) T V \T A ˜ f (y) = inf fa (ξ(a))) ∂ξ = (−y, 0) ∈ Z × Z ,ξ ∈ Z (y ∈ ZT ), ξ ) a∈A ) ( * )  ) V \T A g˜(q) = inf ga (η(a))) η = −δ(q, r), r ∈ Z ,η ∈ Z (q ∈ ZT ), η,r ) a∈A

which are identical to (2.42) and (2.43), respectively, and the claims in Theorem 9.26 reduce to the facts observed in section 2.2.2. Whereas Theorem 9.26 deals with integer-valued functions defined on integer points, similar statements are true for functions of type Z → R and R → R. Note that the conjugacy assertion is missing in the case of Z → R. Theorem 9.27. Assume fa , ga ∈ C[Z → R] for each a ∈ A. For f, g : ZS → R ∪ {+∞}, let f˜, g˜ : ZT → R ∪ {±∞} be the functions induced by G = (V, A; S, T ) according to (9.81) and (9.82), where it is assumed that f˜ > −∞, f˜ ≡ +∞, g˜ > −∞, and g˜ ≡ +∞. (1) f ∈ M[Z → R] =⇒ f˜ ∈ M[Z → R], and f ∈ M [Z → R] =⇒ f˜ ∈ M [Z → R]. (2) g ∈ L[Z → R] =⇒ g˜ ∈ L[Z → R], and g ∈ L [Z → R] =⇒ g˜ ∈ L [Z → R]. Theorem 9.28. Assume fa , ga ∈ C[R → R] for each a ∈ A. For f, g : RS → R ∪ {+∞}, let f˜, g˜ : RT → R ∪ {±∞} be the functions induced by G = (V, A; S, T ) according to ) ( )  ) ˜ fa (ξ(a))) ∂ξ = (x, −y, 0), f (y) = inf f (x) + ) ξ,x a∈A * ξ ∈ RA , (x, −y, 0) ∈ RS × RT × RV \(S∪T ) ( g˜(q) = inf

η,p,r

g(p) +

 a∈A

) ) ) ga (η(a))) η = −δ(p, q, r), )

η ∈ R , (p, q, r) ∈ R × R × R A

S

T

V \(S∪T )

(y ∈ RT ),

* (q ∈ RT ),

where it is assumed that f˜ > −∞, f˜ ≡  +∞, g˜ > −∞, and g˜ ≡ +∞. (1) f ∈ M[R → R] =⇒ f˜ ∈ M[R → R], and f ∈ M [R → R] =⇒ f˜ ∈ M [R → R].

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Chapter 9. Network Flows S S

V , f

z :

T V1 , f1 U f U∗ u0

V2 , f2

T j j j > j > > >

V , f1 2Z f2

Figure 9.5. Bipartite graphs for aggregation and convolution operations.

(2) g ∈ L[R → R] =⇒ g˜ ∈ L[R → R], and g ∈ L [R → R] =⇒ g˜ ∈ L [R → R]. (3) f ∈ M [R → R], g ∈ L [R → R], g = f • , ga = fa • (a ∈ A) =⇒ g˜ = f˜• , where • means the Legendre–Fenchel transformation (3.26).

A number of fundamental operations on M-convex and L-convex functions can be formulated as the transformation by networks, as is partly demonstrated below.

Note 9.29. The M-convexity of the aggregation f U∗ of an M-convex function f (Theorem 6.13 (7)) is proved here as an application of Theorem 9.27. Let V  be a copy of V and consider a bipartite graph G = (S ∪ T, A; S, T ) with S = V  , T = U ∪ {u0 }, and A = {(v  , v) | v ∈ U } ∪ {(v  , u0 ) | v ∈ V \ U }, where v  ∈ V  is the copy of v ∈ V and u0 is a distinguished vertex (see Fig. 9.5 (left)). We regard f as being defined on S and assume that the arc cost functions fa (a ∈ A) are identically zero. The function f˜ induced on T coincides with the aggregation f U∗ . This also means that the aggregation f U∗ can be evaluated by solving an M-convex submodular flow problem. Note 9.30. The M-convexity of the infimal convolution f1 2Z f2 of M-convex functions (Theorem 6.13 (8)) is proved here as an application of Theorem 9.27. Let V1 and V2 be copies of V and consider a bipartite graph G = (S ∪ T, A; S, T ) with S = V1 ∪ V2 , T = V , and A = {(v1 , v) | v ∈ V } ∪ {(v2 , v) | v ∈ V }, where vi ∈ Vi is the copy of v ∈ V for i = 1, 2 (see Fig. 9.5 (right)). We regard fi as being defined on Vi for i = 1, 2 and assume that the arc cost functions fa (a ∈ A) are identically zero. The function f˜ induced on T coincides with the infimal convolution f1 2Z f2 . This also means that the infimal convolution f1 2Z f2 can be evaluated by solving an M-convex submodular flow problem.

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9.6. Network Duality

273 a{1} u{123} * j * aV  u j {23} u0 uV ~a{45} * u{45}j

u{1} u{2} u{3} u{4} u{5}

Figure 9.6. Rooted directed tree for a laminar family.

Note 9.31. An alternative proof of the M -convexity of a laminar convex function (6.34) is given here as an application of Theorem 9.27. Let T be a laminar family of subsets of V , where we may assume that ∅ ∈ / T , V ∈ T , and every singleton set belongs to T . We represent T by a directed tree G = (U, A; S, T ) with root u0 , where U = {uX | X ∈ T } ∪ {u0 }, A = {aX | X ∈ T }, S = {u0 }, T = ˆ denotes {u{v} | v ∈ V }, and ∂ − aX = uX and ∂ + aX = uXˆ for X ∈ T , where X ˆ the smallest member of T that properly contains X (and V = 0 by convention). As an example, the rooted directed tree (arborescence) for V = {1, 2, 3, 4, 5} and T = {{1}, {2}, {3}, {4}, {5}, {2, 3}, {1, 2, 3}, {4, 5}, V } is depicted in Fig. 9.6. We associate the given function fX with arc aX for X ∈ T . The function f˜ on T induced from f = 0 on S by this network coincides with the laminar convex function (6.34), and its M -convexity follows from Theorem 9.27.

9.6.2

Technical Supplements

Proof of Theorem 9.26 It suffices to consider the case of f ∈ M[Z → Z] and g ∈ L[Z → Z]. (1) To prove (M-EXC[Z]) for f˜, we fix y1 , y2 ∈ dom f˜ and u ∈ supp+ (y1 − y2 ) and look for v ∈ supp− (y1 − y2 ) such that58 f˜(y1 ) + f˜(y2 ) ≥ f˜(y1 − χTu + χTv ) + f˜(y2 + χTu − χTv )

(9.84)

by a refinement of the augmenting path argument used in Note 2.19 for the special case of S = ∅. We take ξi ∈ ZA and xi ∈ ZS for i = 1, 2 such that  f˜(yi ) = f (xi ) + fa (ξi (a)), ∂ξi = (xi , −yi , 0) (i = 1, 2). (9.85) a∈A

We search for a kind of augmenting path with respect to the pair (ξ1 , ξ2 ) that yields the desired inequality (9.84). If we are not successful in finding such a path, we modify the flow pair to a new pair with smaller 1 -distance ||ξ1 − ξ2 ||1 , so that we can eventually find an appropriate augmenting path. 58 For

T w ∈ T , χT w is the characteristic vector of {w} in Z .

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Chapter 9. Network Flows

Before giving a formal proof we explain the idea of the proof in a typical situation. Consider the difference in the flows, ξ2 − ξ1 ∈ ZA , for which we have ∂(ξ2 − ξ1 ) = (x2 − x1 , y1 − y2 , 0). Since u ∈ supp+ (∂(ξ2 − ξ1 )), there exists a simple path, say, P1 , compatible with ξ2 − ξ1 that connects u to some vertex v1 in supp− (∂(ξ2 − ξ1 )) = supp+ (x1 − x2 ) ∪ supp− (y1 − y2 ). This is an augmenting path with respect to the pair of flows ξ1 and ξ2 . Suppose that we have the case of v1 ∈ supp+ (x1 − x2 ). By (M-EXC[Z]) for f we obtain u1 ∈ supp− (x1 − x2 ) such that59 f (x1 ) + f (x2 ) ≥ f (x1 − χSv1 + χSu1 ) + f (x2 + χSv1 − χSu1 ). (9.86) Since u1 ∈ supp+ (∂(ξ2 − ξ1 )), there exists a simple path, say, P2 , compatible with ξ2 − ξ1 that connects u1 to some v2 ∈ supp− (∂(ξ2 − ξ1 )) = supp+ (x1 − x2 ) ∪ supp− (y1 − y2 ). Suppose further that v2 ∈ supp− (y1 − y2 ) and P2 is vertex disjoint from P1 . Putting v = v2 we represent the path P1 ∪ P2 by π : A → {0, ±1} such that supp+ (π) ⊆ supp+ (ξ2 − ξ1 ), supp− (π) ⊆ supp− (ξ2 − ξ1 ), and ∂π = (χSu1 − χSv1 , χTu − χTv , 0). For the augmented flows ξ1 = ξ1 + π and ξ2 = ξ2 − π and the new bases x1 = x1 − χSv1 + χSu1 and x2 = x2 + χSv1 − χSu1 , we have ∂ξ1 = (x1 , −(y1 − χTu + χTv ), 0),

∂ξ2 = (x2 , −(y2 + χTu − χTv ), 0)

(9.87)

and fa (ξ1 (a)) + fa (ξ2 (a)) ≥ fa (ξ1 (a)) + fa (ξ2 (a))

(a ∈ A),

(9.88)

since ξ1 (a) < ξ2 (a) ⇒ fa (ξ1 (a)) + fa (ξ2 (a)) ≥ fa (ξ1 (a) + 1) + fa (ξ2 (a) − 1), ξ1 (a) > ξ2 (a) ⇒ fa (ξ1 (a)) + fa (ξ2 (a)) ≥ fa (ξ1 (a) − 1) + fa (ξ2 (a) + 1). By (9.85), (9.86), (9.88), and (9.87), we obtain  [fa (ξ1 (a)) + fa (ξ2 (a))] f˜(y1 ) + f˜(y2 ) = [f (x1 ) + f (x2 )] + a∈A



[f (x1 )

+

f (x2 )]

+



[fa (ξ1 (a)) + fa (ξ2 (a))]

a∈A

≥ f˜(y1 − χTu + χTv ) + f˜(y2 + χTu − χTv ), which shows the inequality (9.84). Having presented the rough idea, we are now in a position to start the proof that works in general. We shall construct a pair of flows ξ1 and ξ2 that satisfy (9.87) for some v ∈ − supp (y1 − y2 ) and x1 , x2 ∈ ZS , and also Φ(ξ1 , ∂ξ1 ) + Φ(ξ2 , ∂ξ2 ) ≤ Φ(ξ1 , ∂ξ1 ) + Φ(ξ2 , ∂ξ2 ), where Φ(ϕ, b) = f (b|S ) +



fa (ϕ(a))

(ϕ ∈ ZA , b ∈ ZV ),

a∈A 59 For

S w ∈ S, χS w is the characteristic vector of {w} in Z .

(9.89)

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275

with b|S denoting the restriction of b to S. To obtain such (ξ1 , ξ2 ) we generate a sequence of tuples (ϕ1 , ϕ2 , b1 , b2 , w) with ϕ1 , ϕ2 ∈ ZA , b1 , b2 ∈ ZV , and w ∈ V such that ϕi (a) ∈ dom fa bi |S ∈ dom f,

(a ∈ A, i = 1, 2), bi |V \(S∪T ) = 0

b1 |T = −(y1 − χTu ),

(i = 1, 2),

b2 |T = −(y2 + χTu ),

b1 = ∂ϕ1 + χw , b2 = ∂ϕ2 − χw , Φ(ϕ1 , b1 ) + Φ(ϕ2 , b2 ) ≤ Φ(ξ1 , ∂ξ1 ) + Φ(ξ2 , ∂ξ2 ),

(9.90) (9.91) (9.92) (9.93) (9.94)

where χw ∈ ZV in (9.93). We call (ϕ1 , ϕ2 , b1 , b2 , w) a flow-boundary tuple and w the frontier vertex. Note that (b1 , b2 ) is determined uniquely from (ϕ1 , ϕ2 , w) by (9.93). We start with (ϕ1 , ϕ2 , b1 , b2 , w) = (ξ1 , ξ2 , ∂ξ1 +χu , ∂ξ2 −χu , u), which obviously satisfies (9.90) to (9.94). If the frontier vertex w is in supp− (y1 − y2 ), we end successfully with (ξ1 , ξ2 , v) = (ϕ1 , ϕ2 , w); note that for w ∈ T we have ∂ϕ1 = (b1 |S , −(y1 − χTu + χTw ), 0) and ∂ϕ2 = (b2 |S , −(y2 + χTu − χTw ), 0). A flow-boundary tuple (ϕ1 , ϕ2 , b1 , b2 , w) is updated to another flow-boundary tuple (ϕ1 , ϕ2 , b1 , b2 , w ) in three ways: (i) flow push, (ii) basis exchange, and (iii) crossover. In all three cases, ||ϕ1 − ϕ2 ||1 ≥ ||ϕ1 − ϕ2 ||1 and the conditions (9.90) to (9.94) are maintained. A flow push augments one unit of flow along an arc a∗ incident to the frontier vertex w. If w = ∂ + a∗ (w is the initial vertex of a∗ ) and ϕ1 (a∗ ) < ϕ2 (a∗ ), the flows are updated as ϕ1 (a∗ ) = ϕ1 (a∗ ) + 1, ϕ2 (a∗ ) = ϕ2 (a∗ ) − 1, and the frontier vertex is changed to w = ∂ − a∗ , the terminal vertex of a∗ . Symmetrically, if w = ∂ − a∗ and ϕ1 (a∗ ) > ϕ2 (a∗ ), the flows and the frontier vertex are updated as ϕ1 (a∗ ) = ϕ1 (a∗ ) − 1, ϕ2 (a∗ ) = ϕ2 (a∗ ) + 1, w = ∂ + a∗ . The boundaries remain unchanged; b1 = b1 and b2 = b2 . See (9.88) for the condition (9.94). A basis exchange applies when w ∈ S and b1 (w) > b2 (w). By (M-EXC[Z]) for f there exists w ∈ S such that b1 (w ) < b2 (w ) and f (b1 |S ) + f (b2 |S ) ≥ f (b1 |S − χSw + χSw ) + f (b2 |S + χSw − χSw ). The frontier vertex is updated to this w and the boundaries to b1 = b1 − χw + χw , b2 = b2 + χw − χw . The flows remain the same: ϕ1 = ϕ1 and ϕ2 = ϕ2 . A crossover updates (ϕ1 , ϕ2 , b1 , b2 , w) with reference to another flow-boundary tuple (ϕ◦1 , ϕ◦2 , b◦1 , b◦2 , w◦ ) with the same frontier vertex w◦ = w. The flow-boundary tuple is updated to (ϕ1 , ϕ2 , b1 , b2 , w ) = (ϕ1 , ϕ◦2 , b1 , b◦2 , w) or (ϕ◦1 , ϕ2 , b◦1 , b2 , w) according to whether Φ(ϕ1 , b1 ) + Φ(ϕ◦2 , b◦2 ) ≤ Φ(ϕ◦1 , b◦1 ) + Φ(ϕ2 , b2 ) or not. The condition (9.94) is maintained, since Φ(ϕ1 , b1 ) + Φ(ϕ2 , b2 ) ≤

1 [Φ(ϕ1 , b1 ) + Φ(ϕ2 , b2 ) + Φ(ϕ◦1 , b◦1 ) + Φ(ϕ◦2 , b◦2 )]. 2

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Under the additional conditions that ϕ1 + ϕ2 = ϕ◦1 + ϕ◦2 , we have

ϕi ∈ [ϕ◦1 ∧ ϕ◦2 , ϕ◦1 ∨ ϕ◦2 ]Z ,

ϕi = ϕ◦i

||ϕ◦1 − ϕ◦2 ||1 − 1 ≥ ||ϕ1 − ϕ2 ||1 .

(i = 1, 2),

(9.95) (9.96)

We use crossovers to avoid cycling in generating the flow-boundary tuples. The generation of flow-boundary tuples consists of stages. Each stage consists of repeated applications of flow push and base exchange, possibly followed by an ap(1) (1) (1) (1) plication of crossover. Denote by (ϕ1 , ϕ2 , b1 , b2 , w(1) ) the flow-boundary tuple (j) (j) (j) (j) at the beginning of a stage and by (ϕ1 , ϕ2 , b1 , b2 , w(j) ) (j = 2, . . . , k) the flowboundary tuples generated so far in the stage. We end the stage if either (a) w(k) ∈ supp− (y1 − y2 ) or (b) w(k) = u = w(1) ; we are done in case (a), whereas in case (b) we go on to the next stage. If w(k) = w(j) for some j with 1 ≤ j ≤ k − 1, we apply (k) (k) (k) (k) (j) (j) (j) (j) crossover to (ϕ1 , ϕ2 , b1 , b2 , w(k) ) with reference to (ϕ1 , ϕ2 , b1 , b2 , w(j) ) (k+1) (k+1) (k+1) (k+1) , ϕ2 , b1 , b2 , w(k+1) ) and to obtain the next flow-boundary tuple (ϕ1 end this stage to go on to the next stage. Otherwise, we apply flow push or base exchange, whichever is applicable, to generate the next flow-boundary tuple (k+1) (k+1) (k+1) (k+1) , ϕ2 , b1 , b2 , w(k+1) ) in the current stage. We prohibit, however, (ϕ1 applying flow push on an arc a∗ right after a flow push on the same arc a∗ (this is possible if |ϕ1 (a∗ ) − ϕ2 (a∗ )| = 1). We also prohibit applying a base exchange with a pair (w, w ) right after a base exchange with (w , w) (this is possible if b1 (w) = b2 (w) + 1 and b1 (w ) = b2 (w ) − 1). Thus, a stage terminates if (a) w(k) ∈ supp− (y1 − y2 ), (b) w(k) = u = w(1) , or (c) a crossover is applied. In case (a) we have successfully found v = w(k) for (9.84). In case (b) we have b1 = ∂ϕ1 + χu , b2 = ∂ϕ2 − χu , and w = u for (k) (k) (k) (k) (ϕ1 , ϕ2 , b1 , b2 , w) = (ϕ1 , ϕ2 , b1 , b2 , w(k) ), just as we had for (ϕ1 , ϕ2 , b1 , b2 , w) = (ξ1 , ξ2 , ∂ξ1 + χu , ∂ξ2 − χu , u) at the beginning of the generation process. In case (c) we obtain a closer pair of flows, with which the next stage starts. The above generation process terminates with a finite number of flow-boundary tuples. This is because (i) the frontier vertices in one stage are distinct and hence the number of flow-boundary tuples generated in one stage is bounded by |V |, (ii) ||ϕ1 − ϕ2 ||1 decreases at least by one at a stage ending in case (c) (note that the conditions in (9.95) are met and (9.96) holds true), and (iii) a stage ending in case (b) must be preceded by a stage ending in case (c). We have thus shown how to construct a desired pair of flows (ξ1 , ξ2 ) by generating flow-boundary tuples starting with (ϕ1 , ϕ2 , b1 , b2 , w) = (ξ1 , ξ2 , ∂ξ1 + χu , ∂ξ2 − χu , u). This completes the proof of (1). (2) Put α = g(p + 1) − g(p), which is independent of p by (TRF[Z]). Since δ(p + 1, q + 1, r + 1) = δ(p, q, r), we have ) * ( )  ) ga (η(a))) η = −δ(p , q + 1, r ) g(p ) + g˜(q + 1) = inf η,p ,r  ) a∈A ) ( * )  ) = inf g(p + 1) + ga (η(a))) η = −δ(p, q, r) = g˜(q) + α, η,p,r ) a∈A

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which shows (TRF[Z]) for g˜. Suppose that g˜(q) is finite for q = q1 , q2 . There exist (η1 , p1 , r1 ) and (η2 , p2 , r2 ) such that  g˜(qi ) = g(pi ) + ga (ηi (a)), ηi = −δ(pi , qi , ri ) (i = 1, 2). a∈A

Here we have g(p1 ) + g(p2 ) ≥ g(p1 ∨ p2 ) + g(p1 ∧ p2 ) by the submodularity (SBF[Z]) of g and ga (η1 (a)) + ga (η2 (a)) ≥ ga (η∨ (a)) + ga (η∧ (a))

(a ∈ A),

with η∨ = −δ(p1 ∨ p2 , q1 ∨ q2 , r1 ∨ r2 ),

η∧ = −δ(p1 ∧ p2 , q1 ∧ q2 , r1 ∧ r2 )

by the convexity of ga (see Note 2.20). The submodularity (SBF[Z]) of g˜ then follows because  [ga (η∨ (a)) + ga (η∧ (a))] g˜(q1 ) + g˜(q2 ) ≥ g(p1 ∨ p2 ) + g(p1 ∧ p2 ) + a∈A

≥ g˜(q1 ∨ q2 ) + g˜(q1 ∧ q2 ). (3) It follows from ∂ξ = (x, −y, 0) and η = −δ(p, q, r) that η, ξ A = − δ(p, q, r), ξ A = − (p, q, r), ∂ξ V = − p, x S + q, y T (cf. (9.21)), whereas the assumed conjugacy implies  f (x) + g(p) ≥ p, x , [fa (ξ(a)) + ga (η(a))] ≥ η, ξ . a∈A

Therefore, we have + f (x) +



,

+

fa (ξ(a)) + g(p) +

a∈A



, ga (η(a)) ≥ q, y ,

a∈A

from which follows the weak duality f˜(y) + g˜(q) ≥ q, y .

(9.97) ∗





Fix y with f˜(y) finite. By Theorem 9.16 (4) there exists (p , q , r ) such that  f˜(y) = q ∗ , y + inf f [−p∗ ] + inf fa [−η ∗ (a)] a∈A ∗







with η = −δ(p , q , r ). This implies   f˜(y) = q ∗ , y − f • (p∗ ) − fa • (η ∗ (a)) = q ∗ , y − g(p∗ ) − ga (η ∗ (a)) a∈A

a∈A

≤ q ∗ , y − g˜(q ∗ ). Combining this with (9.97) shows g˜ = f˜• . The proof of Theorem 9.26 is completed. It is mentioned that (1) follows from (2) and (3) with the aid of the conjugacy theorem (Theorem 8.12).

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Proof of Theorem 9.27 Because the functions are real valued, the infima in (9.81) and (9.82) may not be attained. The proof for Theorem 9.26 can be adapted to this case by introducing ε > 0 and letting ε → 0 as in Notes 2.19 and 2.20. Proof of Theorem 9.28 The augmenting flow argument for (1) suffers from a technical difficulty that the amount of augmenting flow may possibly converge to zero. To circumvent this difficulty we first prove (2) and (3) and then use the conjugacy (Theorem 8.4) to show (1). See Murota–Shioura [152] for details.

Bibliographical Notes The minimum cost flow problem treated in section 9.1 is one of the most fundamental problems in combinatorial optimization. For network flows, Ford–Fulkerson [53] is the classic, whereas Ahuja–Magnanti–Orlin [1] describes recent algorithmic developments; see also Cook–Cunningham–Pulleyblank–Schrijver [26], Du–Pardalos [43], Korte–Vygen [115], Lawler [119], and Nemhauser–Wolsey [167]. Thorough treatments of the network flow problem on the basis of convex analysis can be found in Iri [94] and Rockafellar [178]. The submodular flow problem was introduced by Edmonds–Giles [46] using crossing-submodular functions. The present form avoids crossing-submodular functions on the basis of the fact, due to Fujishige [61], that the base polyhedron defined by a crossing-submodular function can also be described by a submodular function. See Fujishige [65] for other equivalent neoflow problems, such as the independent flow (Fujishige [59]) and the polymatroidal flow (Hassin [87], Lawler–Martel [120], [121]). The M-convex submodular flow problem was introduced by Murota [142]. Section 9.3 is a collection of standard results on the submodular flow problem. Theorems 9.10 and 9.13 are taken from Fujishige [65] (Theorems 5.1 and 5.11, respectively), where the former is ascribed to Frank [56]. The optimality criterion by potentials for the M-convex submodular flow problem was established by Murota [142] for the integer-flow version (Theorem 9.16) and adapted to the real-valued case (Theorem 9.14) with the integrality assertion (Theorem 9.15) in Iwata–Shigeno [105] and Murota [147]. The optimality criterion by negative cycles (Theorem 9.20) was established by Murota [140], [142] for the integer-flow version and adapted to the real-valued case (Theorem 9.18) in Murota–Shioura [152]. Theorem 9.22 is in [140], [142]. Proposition 9.23 is a reformulation of the unique-max lemma due to Murota [135]. Proposition 9.25 is also from [135]; the proof technique using (9.80) originated in Fujishige [59]. Transformation by networks was found first for M-convex functions f ∈ M[Z → R] by Murota [137]; the proof given in section 9.6.2 is due to Shioura [188], [189]. Transformation of L-convex functions is stated explicitly in Murota [145]. The extension to polyhedral M-convex and L-convex functions is made in Murota–Shioura

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[152]. Theorems 9.26, 9.27, and 9.28 are explicit in Murota [147]. Induction of matroids through graphs is due to Perfect [171] (for bipartite graphs) and Brualdi [21]; see also Schrijver [183], Welsh [211], and White [213]. The alternative proof of the M -convexity of a laminar convex function described in Note 9.31 is communicated by A. Shioura.

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Chapter 10

Algorithms

Algorithmic aspects of M-convex and L-convex functions are discussed in this chapter. Three fundamental optimization problems tractable by efficient algorithms are (i) M-convex function minimization, which is a nonlinear extension of the minimumweight base problem for matroids; (ii) L-convex function minimization, which includes submodular set function minimization as a special case; and (iii) minimization/maximization in the Fenchel-type min-max duality, which is equivalent to the M-convex submodular flow problem.

10.1

Minimization of M-Convex Functions

Four kinds of algorithms for M-convex function minimization are described: the steepest descent algorithm, the steepest descent scaling algorithm, the domain reduction algorithm, and the domain reduction scaling algorithm. Throughout this section we assume that f : ZV → R ∪ {+∞} is an M-convex function, n = |V |, and F is an upper bound on the time to evaluate f .

10.1.1

Steepest Descent Algorithm

The local characterization of global minimality for M-convex functions (Theorem 6.26) immediately suggests the following algorithm of steepest descent type. Steepest descent algorithm for an M-convex function f ∈ M[Z → R] S0: Find a vector x ∈ dom f . S1: Find u, v ∈ V (u = v) that minimize f (x − χu + χv ). S2: If f (x) ≤ f (x − χu + χv ), then stop (x is a minimizer of f ). S3: Set x := x − χu + χv and go to S1. Step S1 can be done with n2 evaluations of function f . At the termination of the algorithm in step S2, x is a global minimizer by Theorem 6.26 (M-optimality criterion). The function value f decreases monotonically with iterations. This 281

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property alone does not ensure finite termination in general, although it does if f is integer valued and bounded from below. Let us derive an upper bound on the number of iterations by considering the distance to the optimal solution rather than the function value. Proposition 10.1. If f has a unique minimizer, say, x∗ , the number of iterations in the steepest descent algorithm is bounded by ||x◦ − x∗ ||1 /2, where x◦ denotes the initial vector found in step S0. Proof. Put x = x − χu + χv in step S2. By Theorem 6.28 (M-minimizer cut), we have x∗ (u) ≤ x(u) − 1 = x (u) and x∗ (v) ≥ x(v) + 1 = x (v), which implies ||x − x∗ ||1 = ||x − x∗ ||1 − 2. Note that ||x◦ − x∗ ||1 is an even integer. When given an M-convex function f , which may have multiple minimizers, we consider a perturbation of f so that we can use Proposition 10.1. Assume now that the effective domain is bounded and denote its 1 -size by K1 = max{||x − y||1 | x, y ∈ dom f }.

(10.1)

We arbitrarily fix a bijection ϕ : V → {1, 2, . . . , n} to represent an ordering of the elements of V , put vi = ϕ−1 (i) for i = 1, . . . , n, and define a vector p ∈ RV by p(vi ) = εi for i = 1, . . . , n, where ε > 0. The function fε = f [p] is M-convex by Theorem 6.13 (3) and, for a sufficiently small ε, it has a unique minimizer that is also a minimizer of f . Suppose that the steepest descent algorithm is applied to the n perturbed function fε . Since fε (x − χu + χv ) = f (x − χu + χv ) + i=1 εi x(vi ) − ϕ(u) ϕ(v) ε +ε , this amounts to employing a tie-breaking rule: take (u, v) that lexicographically minimizes Φ(u, v), where

 Φ(u, v) =

(−1, ϕ(u), −ϕ(v)) (+1, −ϕ(v), ϕ(u))

(10.2)

if ϕ(u) < ϕ(v), if ϕ(u) > ϕ(v),

in the case of multiple candidates in step S1 of the steepest descent algorithm applied to f . With this tie-breaking rule we have the following complexity bound. Proposition 10.2. For an M-convex function f with finite K1 in (10.1), the number of iterations in the steepest descent algorithm with tie-breaking rule (10.2) is bounded by K1 /2. Hence, if a vector in dom f is given, the algorithm finds a minimizer of f in O(F · n2 K1 ) time. By Theorem 6.76 (quasi M-optimality criterion) and Theorem 6.77 (quasi Mminimizer cut), the steepest descent algorithm can also be used for minimizing quasi M-convex functions satisfying (SSQM = ). Note 10.3. For integrally convex functions we have the local optimality criterion for global optimality (Theorem 3.21). This naturally suggests the following.

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Steepest descent algorithm for an integrally convex function f S0: Find a vector x ∈ dom f . S1: Find disjoint Y, Z ⊆ V that minimize f (x − χY + χZ ). S2: If f (x) ≤ f (x − χY + χZ ), then stop (x is a minimizer of f ). S3: Set x := x − χY + χZ and go to S1. The steepest descent algorithm for M-convex functions is a special case of this. It is emphasized that no efficient algorithm for step S1 is available for general integrally convex functions.

10.1.2

Steepest Descent Scaling Algorithm

Scaling is one of the fundamental general techniques in designing efficient algorithms. The proximity theorem for M-convex functions leads us to the following steepest descent scaling algorithm for M-convex function minimization. We assume that the effective domain is bounded and denote its ∞ -size by K∞ = max{||x − y||∞ | x, y ∈ dom f }.

(10.3)

Steepest descent scaling algorithm for an M-convex function f ∈ M[Z → R] S0: Find a vector x ∈ dom f , and set α := 2log2 (K∞ /4n) , B := dom f . S1: Find an integer  vector y that locally minimizes f (x + αy) (x + αy ∈ B), f˜(y) = +∞ (x + αy ∈ / B) ˜ ˜ in the sense of f (y) ≤ f (y − χu + χv ) (∀ u, v ∈ V ) by the steepest descent algorithm of section 10.1.1 with initial vector 0 and set x := x + αy. S2: If α = 1, then stop (x is a minimizer of f ). S3: Set B := B ∩ {y ∈ ZV | ||y − x||∞ ≤ (n − 1)(α − 1)} and α := α/2 and go to S1. By the M-proximity theorem (Theorem 6.37 (1)), the set B always contains a global minimizer of f and, at the termination of the algorithm in step S2, x is a global minimizer by the M-optimality criterion (Theorem 6.26 (1)). The number of iterations is bounded by log2 (K∞ /4n) . Some remarks are in order concerning step S1. If the function f is such that the scaled function f˜ remains M-convex, step S1 can be done in O(F · n4 ) time by the steepest descent algorithm with tie-breaking rule (10.2). This time bound follows from Proposition 10.2 with K1 ≤ 4n2 . For a general f , however, f˜ is not necessarily M-convex (see Note 6.18) and no polynomial bound for step S1 is guaranteed, although we have an obvious exponential time bound O(F · (4n)n n2 ). On the basis of Theorem 6.76 (quasi M-optimality criterion) and Theorem 6.78 (quasi M-proximity theorem), the steepest descent scaling algorithm can be adapted to the minimization of quasi M-convex functions satisfying (SSQM = ).

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10.1.3

Chapter 10. Algorithms

Domain Reduction Algorithm

The domain reduction algorithm is a kind of bisection method that searches for the minimum of an M-convex function by generating a sequence of nested subsets of the domain on the basis of the M-minimizer cut theorem (Theorem 6.28). For an M-convex function with bounded effective domain, the algorithm finds a minimizer in time polynomial in n and log2 K∞ , where K∞ is defined by (10.3). We introduce the following notations for a bounded nonempty set B ⊆ ZV : (v ∈ V ), B (v) = min y(v), uB (v) = max y(v) y∈B y∈B        1 1 1 1 B + 1 − ◦B = 1− u◦B = B + uB , uB , n n n n B ◦ = {y ∈ B | ◦B ≤ y ≤ u◦B }. The set B ◦ is intended to represent the central part of B, i.e., the set of vectors of B lying away from the boundary. The set B ◦ is nonempty if B is M-convex; see Proposition 10.6 below. Domain reduction algorithm for an M-convex function f ∈ M[Z → R] S0: Set B := dom f . S1: Find a vector x ∈ B ◦ . S2: Find u, v ∈ V (u = v) that minimize f (x − χu + χv ). S3: If f (x) ≤ f (x − χu + χv ), then stop (x is a minimizer of f ). S4: Set B := B ∩ {y ∈ ZV | y(u) ≤ x(u) − 1, y(v) ≥ x(v) + 1} and go to S1. The vector x ∈ B ◦ in step S1 can be found with O(n2 log2 K∞ ) evaluations of f by the procedure to be described below. The set B forms a decreasing sequence of M-convex sets, which contain a minimizer of f because of the M-minimizer cut theorem (Theorem 6.28). Since x is taken from the central part of B, uB (w)−B (w) for w ∈ {u, v} decreases with a factor of (1 − n1 ) and hence the number of iterations is bounded by O(n2 log2 K∞ ). The above algorithm, therefore, finds a minimizer of f with O(n4 (log2 K∞ )2 ) evaluations of f provided that it is given a vector in dom f . Proposition 10.4. If a vector in dom f is given, the domain reduction algorithm finds a minimizer of an M-convex function f in O(F · n4 (log2 K∞ )2 ) time. It remains to show how to find x ∈ B ◦ in step S1 when given a vector of B. For a vector y of an M-convex set B and two distinct elements u, v of V , the exchange capacity is defined by cB (y, v, u) = max{α ∈ Z | y + α(χv − χu ) ∈ B},

(10.4)

which is a nonnegative integer representing the distance from y to the boundary of B in the direction of χv − χu . In the domain reduction algorithm, B is always an M-convex set and the exchange capacity can be computed by a binary search with log2 K∞ evaluations of f . For x ∈ B we define V ◦ (x) = {v ∈ V | ◦B (v) ≤ x(v) ≤ u◦B (v)}

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with an obvious observation that V ◦ (x) = V if and only if x ∈ B ◦ . If V ◦ (x) = V , we can modify x to x ∈ B with the property |V ◦ (x )| ≥ ◦ |V (x)| + 1 as follows. Take any u ∈ V \ V ◦ (x) and assume x(u) > u◦B (u); the other case with x(u) < ◦B (u) can be treated in a similar manner. Putting {v1 , v2 , . . . , vn−1 } = V \ {u} and x0 = x, we define a sequence x1 , x2 , . . . , xn−1 by xi = xi−1 + αi (χvi − χu ) with ⎧ if u◦B (vi ) ≤ xi−1 (vi ), ⎨ 0 ◦ min[cB (xi−1 , vi , u), xi−1 (u) − uB (u), αi = ⎩ otherwise u◦B (vi ) − xi−1 (vi )] for i = 1, 2, . . . , n − 1 and put x = xn−1 . Proposition 10.5. V ◦ (x ) ⊇ V ◦ (x) ∪ {u}. Proof. The inclusion V ◦ (x ) ⊇ V ◦ (x) is obvious. To prove x (u) = u◦B (u) by contradiction, suppose x (u) > u◦B (u) and take x∗ ∈ B ◦ , where B ◦ = ∅ by Proposition 10.6 below. Since u ∈ supp+ (x − x∗ ), it follows from (M-EXC[Z]) that x ≡ x + χvi − χu ∈ B and x (vi ) < x∗ (vi ) ≤ u◦B (vi ) for some vi . Since vi ∈ supp+ (x − xi ) and supp− (x − xi ) = {u}, (M-EXC[Z]) implies xi + χvi − χu ∈ B. But this contradicts the definition of xi . The modification of x to x described above can be done with n evaluations of the exchange capacity. Repeating such modifications at most n times we arrive at x with V ◦ (x) = V . Thus, given a vector in B, we can find x ∈ B ◦ with at most n2 evaluations of the exchange capacity. We finally prove the nonemptiness of B ◦ . Proposition 10.6. B ◦ = ∅ if B is an M-convex set. Proof. Let ρ ∈ S[Z] be the submodular function satisfying B = B(ρ) ∩ ZV and ρˆ be its Lov´asz extension. Then we have B (v) = ρ(V )− ρ(V \ {v}) and uB (v) = ρ(v). We can assert the nonemptiness of B ◦ by establishing (i) ◦B (X) ≤ ρ(X) (X ⊆ V ),

(ii) u◦B (X) ≥ ρ(V ) − ρ(V \ X) (X ⊆ V )

(see Theorem 3.8 in Fujishige [65]). We prove (i) here; a similar argument works for (ii). Fix X ⊆ V and put p = χX , pv = 1 − χv (v ∈ X), and k = |X|. It follows from   k1 = p + p v , χv = p + pu − (k − 1)1 (v ∈ X) v∈X

u∈X\{v}



that kρ(V ) = ρˆ(k1) = ρˆ(p + ⎛  ρ(v) = ρˆ(χv ) = ρˆ ⎝p +

u∈X\{v}

v∈X

pv ) and ⎞



pu − (k − 1)1⎠ = ρˆ ⎝p +

 u∈X\{v}

⎞ pu ⎠−(k−1)ρ(V ).

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With these identities we see   1 1 B (X) + uB (X) ≤ ρ(X) 1− n n   ρ(v) ≤ nρ(X) + (n − 1) ρ(V \ {v}) ⇔ (n − 1)kρ(V ) + 8 ⇔ (n − k)ˆ ρ p+

v∈X

 v∈X

pv

9 +





ρˆ ⎝p +

v∈X

The last inequality holds true since 9 8   pv ≤ ρˆ(p) + ρˆ(pv ), ρˆ p + v∈X

v∈X



v∈X



pu ⎠ ≤ nˆ ρ(p) + (n − 1)

ρˆ ⎝p +



ρˆ(pv ).

v∈X

u∈X\{v}





⎞ pu ⎠ ≤ ρˆ(p) +

u∈X\{v}



ρˆ(pu )

u∈X\{v}

by the positive homogeneity and convexity of ρˆ. Hence follows ◦B (X) ≤ ρ(X). By Theorem 6.77 (quasi M-minimizer cut), the domain reduction algorithm can also be used for minimizing quasi M-convex functions satisfying (SSQM = ) provided that the effective domain is a bounded M-convex set.

10.1.4

Domain Reduction Scaling Algorithm

We present here the domain reduction scaling algorithm for M-convex function minimization, a combination of the idea of the domain reduction algorithm of section 10.1.3 with a scaling technique based on the theorem of M-minimizer cut with scaling (Theorem 6.39). The algorithm works with a pair (x, ) of integer vectors, where x is the current solution and  is a lower bound for an optimal solution. Specifically, two conditions x ∈ S() ∩ dom f,

S() ∩ arg min f = ∅

(10.5)

are maintained, where S() = {y ∈ ZV | y ≥ }. The algorithm consists of scaling phases parametrized (or labeled) by a nonnegative integer α, called the scaling factor, which is initially set to be sufficiently large and is decreased until it reaches unity. In each scaling phase with a fixed α, the pair (x, ) is modified so that it satisfies an additional condition  ≤ x ≤  + (n − 1)(α − 1)1.

(10.6)

At the end of the algorithm we have α = 1 and hence x =  by (10.6). This means S() ∩ dom f = {x}, since x(V ) = y(V ) for any y ∈ dom f by Proposition 6.1 and since x(V ) < y(V ) for any y ∈ S() distinct from x. Furthermore, we see from the second condition in (10.5) that x is a minimizer of f , since {x} = S() ∩ dom f ⊇ S() ∩ arg min f = ∅. The outline of the algorithm reads as follows, where step S1 for the α-scaling phase is described later and K∞ is defined in (10.3).

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Domain reduction scaling algorithm for an M-convex function f ∈ M[Z → R] S0: Find a vector x ∈ dom f and set  := x − K∞ 1, α := 2log2 (K∞ /2n) . S1: Modify (x, ) to meet (10.6) (α-scaling phase). S2: If α = 1, then stop (x is a minimizer of f ). S3: Set α := α/2 and go to S1. The α-scaling phase is now described. In view of (10.6) we employ a subset V • of V such that x(w) ≤ (w) + (n − 1)(α − 1)

(∀ w ∈ V \ V • ).

(10.7)

Initially, V • is set to V and then decreases monotonically to the empty set. α-scaling phase for (x, , α) S0: Set V • := V . S1: If V • = ∅, then output (x, ) and stop. S2: Take any u ∈ V • . S3: Find v ∈ V that minimizes f (x + α(χv − χu )). S4: If v = u or x(u) − α < (u), then set (u) := max[(u), x(u) − (n − 1)(α − 1)] and V • := V • \ {u} and go to S1. S5: Otherwise, set (v) := max[(v), x(v) + α − (n − 1)(α − 1)], x := x + α(χv − χu ) and V • := V • \ {v} and go to S1. As is easily seen, the first condition in (10.5) is maintained in steps S4 and S5. The second condition in (10.5) is also maintained by virtue of Theorem 6.39 (Mminimizer cut with scaling). The subset V • is nonincreasing, although it may be that v ∈ / V • in step S5, and then the operation V • := V • \ {v} is void and V • remains unchanged. Denote the initial value of (x, ) by (x◦ , ◦ ). For each w ∈ V the value of x(w) is decreased at most (x◦ (w) − ◦ (w))/α times before it is deleted from V • . Hence, the number of iterations in the α-scaling phase is bounded by ||x◦ − ◦ ||1 /α. In particular, the α-scaling phase terminates with V • = ∅. The time complexity of the domain reduction scaling algorithm is given as follows, where it is assumed for simplicity that K∞ is known. Proposition 10.7. If a vector in dom f is given, the domain reduction scaling algorithm finds a minimizer of an M-convex function f in O(F · n3 log2 (K∞ /n)) time. Proof. At the beginning of the α-scaling phase we have ||x◦ −◦ ||1 ≤ n(n−1)(2α−1) by (10.6). Since step S3 in the α-scaling phase can be done with n evaluations of f , the α-scaling phase terminates in O(F · n3 ) time. The number of scaling phases is equal to log2 (K∞ /2n) . On the basis of Theorem 6.79 (quasi M-minimizer cut with scaling), the domain reduction scaling algorithm can be adapted to the minimization of quasi M-convex functions satisfying (SSQM = ) provided that the effective domain is a bounded M-convex set.

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10.2

Chapter 10. Algorithms

Minimization of Submodular Set Functions

The minimization of submodular set functions is one of the most fundamental problems in combinatorial optimization. In this section we deal with algorithms for minimizing submodular set functions, which we will use as an essential component in algorithms for L-convex functions.

10.2.1

Basic Framework

Let ρ : 2V → R be a submodular set function,60 where ρ(∅) = 0 and n = |V |. In discussing the efficiency or complexity of algorithms it is customary to categorize them into finite, pseudopolynomial, weakly polynomial and strongly polynomial algorithms. For our problem of minimizing ρ, a finite algorithm is trivial; we may evaluate ρ(X) for all subsets X to find the minimum. This takes O(F · 2n ) time, where F is an upper bound on the time to evaluate ρ. The complexity of an algorithm may depend on the complexity or size of ρ; if ρ is integer valued, M = max |ρ(X)|

(10.8)

X⊆V

often serves as a measure of the size of ρ. An algorithm for minimizing ρ is said to be (i) pseudopolynomial , (ii) weakly polynomial , or (iii) strongly polynomial , according as the total number of evaluations of ρ as well as other arithmetic operations involved is bounded by a polynomial in (i) n and M , (ii) n and log2 M , or (iii) n alone. Our objective in this section is to describe two strongly polynomial algorithms for minimizing ρ. Let B(ρ) = {x ∈ RV | x(X) ≤ ρ(X) (∀ X ⊂ V ), x(V ) = ρ(V )}

(10.9)

be the base polyhedron associated with ρ. Recall that a point in B(ρ) is called a base and an extreme point of B(ρ) is an extreme base. For any base x and any subset X we obviously have x− (V ) ≤ x(X) ≤ ρ(X),

(10.10)

where x− is the vector in RV defined by x− (v) = min(0, x(v)) for v ∈ V . The inequalities are tight for some x and X, as follows. Proposition 10.8. For a submodular set function ρ : 2V → R, we have max{x− (V ) | x ∈ B(ρ)} = min{ρ(X) | X ⊆ V }.

(10.11)

If ρ is integer valued, the maximizer x can be chosen to be an integer vector. Proof. Although this is an easy consequence of Edmonds’s intersection theorem (Theorem 4.18) with ρ1 = ρ and ρ2 = 0, a direct proof is given here. Let x be a 60 Note that we assume ρ to be finite valued for all subsets. An adaptation to the general case ρ : 2V → R ∪ {+∞} is explained in Note 10.14.

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maximizer on the left-hand side. For any u ∈ supp− (x) and v ∈ supp+ (x), there exists5a subset X4 uv such that u ∈ Xuv ⊆ V \ {v} and x(Xuv ) = ρ(Xuv ). Put X = u∈supp− (x) v∈supp+ (x) Xuv . We have x(X) = ρ(X) by (4.23) and x− (V ) = x(X) since supp− (x) ⊆ X and supp+ (x) ⊆ V \ X. The integrality assertion can be established by the same argument starting with an integral base x that maximizes x− (V ) over all integral bases. The min-max relation (10.11) shows that we can demonstrate the optimality of a subset X by finding a base x with x− (V ) = ρ(X). But how can we verify that a vector x belongs to B(ρ)? By definition, x ∈ B(ρ) if and only if minX (ρ(X) − x(X)) = ρ(V ) − x(V ) = 0. Thus, testing for membership in B(ρ) for an arbitrary x seems to need a submodular function minimization procedure. To circumvent this difficulty, we recall from Note 4.10 that we can generate an extreme base by the greedy algorithm. Let L = (v1 , v2 , . . . , vn ) be a linear ordering of V and define L(vj ) = {v1 , v2 , . . . , vj } for j = 1, . . . , n. Then the extreme base y associated with L is given by y(v) = ρ(L(v)) − ρ(L(v) \ {v})

(v ∈ V ).

(10.12)

Any base x can be represented as a convex combination of a number of extreme bases, say, {yi | i ∈ I}, as 9 8   x= λi yi λi = 1, λi > 0 (i ∈ I) , (10.13) with i∈I

i∈I

where we may assume |I| ≤ n by the Carath´eodory theorem. Combining (10.12) and (10.13) shows that any base can be represented by a list of linear orderings {Li | i ∈ I} (that generate {yi | i ∈ I}) and coefficients of the convex combination {λi | i ∈ I}. With this representation of x we can be sure that x is a member of B(ρ). For u, v ∈ V and i ∈ I we use the notation u ≺i v ⇐⇒ u precedes v in Li , (u, v]≺i = {w | u ≺i w (i v},

(10.14) (10.15)

where w (i v means w ≺i v or w = v. The following proposition gives a sufficient condition for optimality in terms of the linear orderings {Li | i ∈ I}. Proposition 10.9. Let x be a base represented as (10.13) with {(Li , λi ) | i ∈ I} and W be a subset of V . (1) If supp− (x) ⊆ W and supp+ (x) ⊆ V \ W , then x− (V ) = x(W ). (2) If u ≺i v for every u ∈ W , v ∈ V \ W , and i ∈ I, then x(W ) = ρ(W ). (3) If the conditions in (1) and (2) are satisfied, x and W are optimal in (10.11). Proof. (1) is obvious. By (10.12) the condition in (2) implies ρ(W ) = yi (W ) for every i ∈ I. Then x(W ) = i∈I λi yi (W ) = ρ(W ). (3) follows from (1), (2), and (10.10).

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290

Chapter 10. Algorithms We are thus led to a basic algorithmic framework:

1. We maintain (and update) a number of linear orderings {Li | i ∈ I}, together with the associated extreme bases {yi | i ∈ I} and the coefficients of convex combination {λi | i ∈ I}, to represent a base x. 2. We terminate when the conditions in Proposition 10.9 are satisfied. Two strongly polynomial algorithms using this framework are described in subsequent sections. Note 10.10. Here is a brief historical account of submodular function minimization. Its importance seems to have been recognized around 1970 by J. Edmonds [44] and others. The first polynomial algorithm was given by M. Gr¨otschel, L. Lov´ asz, and A. Schrijver—weakly polynomial in 1981 [82], and strongly polynomial in 1988 [83]. These algorithms, however, are based on the ellipsoid method and, as such, are not so much combinatorial as geometric. Efforts for a combinatorial polynomial algorithm have been continued with major contributions made by W. H. Cunningham and others [15], [28], [29], who showed the basic framework above as well as a combinatorial pseudopolynomial algorithm for submodular function minimization. In 1994, extending the min-cut algorithm of Nagamochi–Ibaraki [163], M. Queyranne [172] came up with a combinatorial algorithm for symmetric submodular function minimization, which is to minimize over nonempty proper subsets a submodular function ρ such that ρ(X) = ρ(V \ X) for all X ⊆ V . Combinatorial strongly polynomial algorithms for general submodular functions were found, independently, in the summer of 1999 by two groups, S. Iwata, L. Fleischer, and S. Fujishige [101], [102] and A. Schrijver [182]. Both of these follow Cunningham’s framework, but they are significantly distinct in technical aspects. Subsequently, a new problem was recognized by Schrijver. These two algorithms are certainly combinatorial, but they rely on arithmetic operations (division, in particular) in computing the coefficients of convex combination in (10.13). The question posed by Schrijver is as follows: Is it possible to design a fully combinatorial strongly polynomial algorithm that is free from division and relies only on addition, subtraction, and comparison? This was answered in the affirmative by Iwata [99] in the fall of 2000. Note 10.11. Because the minimizers form a ring family (see Note 4.8), there exists a unique minimal minimizer as well as a unique maximal minimizer of a submodular V set function ρ : 2 → R. Given an optimal base x with the representation x = i∈I λi yi in (10.13), we can compute the minimal and maximal minimizers in strongly polynomial time as follows. With the notation D(x) for the family of tight sets at x, introduced in (4.22) of Note 4.9, we have a representation arg min ρ = {X ∈ D(x) | supp− (x) ⊆ X ⊆ V \ supp+ (x)}

(10.16)

for the family of the minimizers of ρ. Noting that D(x) is a ring family, let Gx = (V, Ax ) be the directed graph associated with D(x), as defined by (4.20) in Note

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4.7. This is equivalent to saying that (u, v) ∈ Ax if and only if v ∈ dep(x, u), where dep(x, u) means the smallest tight set at x that contains u. Then the minimal minimizer can be identified as the set of vertices reachable from supp− (x) in Gx and the maximal minimizer as the complement of the set of vertices reachable to supp+ (x) in Gx . Moreover, the graph Gx enables us to enumerate all the minimizers. By (10.13) we have D(x) =

2

D(yi ),

Ax =

i∈I

:

Ayi .

i∈I

Since each yi is an extreme base, we can easily compute dep(yi , u). For each u ∈ V , we start with D := V and update D to D \ {v} as long as D \ {v} ∈ D(yi ) for some v ∈ D \ {u}; we obtain D = dep(yi , u) at the termination. We can thus compute dep(yi , u) with O(n2 ) evaluations of ρ. The number of function evaluations can be reduced to O(n) if a linear ordering Li = (v1 , v2 , . . . , vn ) generating yi is also available; assuming u = vk , start with D = {v1 , v2 , . . . , vk−1 } and, for j = k − 1, k − 2, . . . , 1, update D to D \ {vj } if D \ {vj } ∈ D(yi ). Therefore, the graph Gx can be constructed with O(n3 |I|) or O(n2 |I|) evaluations of function ρ, where it is reasonable to assume |I| ≤ n. Note 10.12. The minimal minimizer of a submodular set function ρ : 2V → R can also be computed using any submodular function minimization algorithm n + 1 times. Let X be a minimizer of ρ. For each v ∈ X, compute a minimizer Yv of a submodular set function ρv : 2X\{v} → R, the restriction of ρ to X \ {v}, defined by ρv (Y ) = ρ(Y ) for Y ⊆ X \ {v}. Then the minimal minimizer of ρ is given by {v ∈ X | ρ(Yv ) > ρ(X)}, since ρ(Yv ) > ρ(X) if v is contained in the minimal minimizer and ρ(Yv ) = ρ(X) if not. The maximal minimizer can be computed similarly. Note 10.13. An alternative way to find the minimal minimizer of a submodular set function ρ : 2V → R is to introduce a penalty term to represent the size of a subset and to minimize a modified submodular set function ρ˜(X) = ρ(X) + ε|X| with a sufficiently small positive parameter ε. If ρ is integer valued, ε = 1/(n + 1) is a valid choice. The maximal minimizer can be computed similarly. Note 10.14. It has been assumed that the submodular function ρ is finite valued for all subsets. This assumption is not restrictive but for convenience of description. Let ρ ∈ S[R] be a submodular set function on V taking values in R ∪{+∞}; D = dom ρ is a ring family with {∅, V } ⊆ D. For v ∈ V we denote by Mv the smallest member of D containing v and by Nv the largest member of D not containing v. For X ⊆ V we denote by X the smallest member of D including X. We assume that we can compute Mv for each v ∈ V efficiently, say, in time polynomial in n. Then we can compute X and Nv by X=

: v∈X

Mv ,

Nv =

: u:v ∈M / u

Mu .

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Let us assume, without loss of generality, that the length of a maximal chain of D is equal to n = |V |. Consider now a set function ρ : 2V → R defined by ρ(X) = ρ(X) + c(X) − c(X)

(X ⊆ V ),

with c ∈ RV given by c(v) = max(0, ρ(Nv ) − ρ(Nv ∪ {v}))

(v ∈ V ).

As is shown below, (i) ρ is a finite-valued submodular function and (ii) X ∈ arg min ρ implies X ∈ arg min ρ. Thus we can minimize ρ via the minimization of ρ. Proof of (i): ρ is obviously finite valued. If X ∈ D, v ∈ / X, and Y = X ∪ {v} ∈ D, we have Y ∪ Nv = Nv ∪ {v} and Y ∩ Nv = X and, therefore, ρ(X) = ρ(Y ∩ Nv ) ≤ ρ(Y ) + ρ(Nv ) − ρ(Y ∪ Nv ) ≤ ρ(Y ) + c(v). This means that μ(X) = ρ(X)+c(X) is nondecreasing on D. Putting μ(X) = μ(X) for X ⊆ V and noting X ∪ Y = X ∪ Y and X ∩ Y ⊇ X ∩ Y , we see μ(X) + μ(Y ) = μ(X) + μ(Y ) ≥ μ(X ∪ Y ) + μ(X ∩ Y ) ≥ μ(X ∪ Y ) + μ(X ∩ Y ) = μ(X ∪ Y ) + μ(X ∩ Y ), which shows the submodularity of μ and hence that of ρ. Proof of (ii): For any Y ∈ D we have ρ(X) ≤ ρ(X) + c(X) − c(X) = ρ(X) ≤ ρ(Y ) = ρ(Y ). Note 10.15. The method in Note 10.14 can be used to minimize a submodular function defined on a (finite) distributive lattice. By a lattice we mean a tuple (S, ∨, ∧) of a nonempty set S and two binary operations ∨ and ∧ on S such that a ∨ a = a,

a ∧ a = a;

a ∨ b = b ∨ a,

a ∧ b = b ∧ a;

a ∨ (b ∨ c) = (a ∨ b) ∨ c, a ∧ (b ∧ c) = (a ∧ b) ∧ c; a ∧ (a ∨ b) = a, a ∨ (a ∧ b) = a for any a, b, c ∈ S. A lattice (S, ∨, ∧) is a distributive lattice if, in addition, the distributive law a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c),

a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

holds true. A ring family D is a typical distributive lattice, where (S, ∨, ∧) = (D, ∪, ∩). The converse is essentially true: any distributive lattice (S, ∨, ∧) can be represented in the form of a ring family (Birkhoff ’s representation theorem). The size of the underlying set of the ring family is equal to the length of a maximal chain of S. A function ρ : S → R defined on a distributive lattice (S, ∨, ∧) is said to be submodular if it satisfies ρ(a) + ρ(b) ≥ ρ(a ∨ b) + ρ(a ∧ b)

(a, b ∈ S).

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Thus, with an appropriate representation of (S, ∨, ∧), a submodular function ρ on S can be minimized by using the method in Note 10.14. Note 10.16. Maximizing a submodular set function is a difficult task in general. It is known that no polynomial algorithm exists for it (and this statement is independent of the P = NP conjecture); see Jensen–Korte [106] and Lov´asz [122], [123]. In this context, M -concave functions on {0, 1}-vectors form a tractable subclass of submodular set functions. Recall that an M -concave function is submodular (Theorem 6.19) and that it can be maximized efficiently by algorithms in section 10.1.

10.2.2

Schrijver’s Algorithm

We explain here Schrijver’s strongly polynomial algorithm for submodular function minimization. This algorithm achieves strong polynomiality using a distance labeling with an ingenious lexicographic rule. Following the basic framework introduced in section 10.2.1, the algorithm employs the representation of a base in terms of a convex combination of extreme bases associated with linear orderings. Given {(Li , λi ) | i ∈ I}, the algorithm constructs a directed graph G = (V, A) with arc set : {(u, v) | u ≺i v} (10.17) A= i∈I

(see (10.14) for the notation ≺i ) and searches for a directed path from P = supp+ (x) to N = supp− (x). If there is no such path, the algorithm terminates by setting W to be the set of vertices reachable to N . Then W and x satisfy all the conditions in Proposition 10.9, and hence W is a minimizer of ρ and x is a maximizer of x− (V ). Otherwise, it modifies {(Li , λi ) | i ∈ I} with reference to a path from P to N . Schrijver’s algorithm for submodular function minimization S0: Take any linear ordering L1 and set I := {1}, λ1 := 1. I} by (10.17). S1: Construct the graph G = (V, A) for {(Li , λi ) | i ∈ S2: Set P := supp+ (x), N := supp− (x) for base x = i∈I λi yi in (10.13). S3: If there exists no directed path from P to N in G, let W be the set of vertices reachable to N and stop (W is a minimizer of ρ). S4: Update {(Li , λi ) | i ∈ I} and go to S1. We now describe the concrete procedure for step S4, where a directed path exists from P to N in G. Let d(v) denote the distance (= minimum number of arcs in a directed path) in G from P to v. We choose s, t ∈ V as follows (a lexicographic rule). We fix a linear ordering ≺0 of elements of V ; this is independent of the linear orderings Li . Let t be the element in N reachable from P with d(v) maximum; in the case of multiple candidates, choose the largest with respect to ≺0 . Let s be the element with (s, t) ∈ A, d(s) = d(t)−1; in the case of multiple candidates, choose the largest with respect to ≺0 . Let α be the maximum of |(s, t]≺i | over i ∈ I and let k ∈ I be such that |(s, t]≺k | = α.

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Index the elements of V so that Lk = (v1 , . . . , vn ) and assume vp = s. Then we have vp+α = t and (s, t]≺k = {vp+1 , . . . , vp+α }. For j = 1, . . . , α, consider a linear ordering s L•j :

t





v1 · · · vp−1 | vp+j | vp · · · vp+j−1 | | vp+j+1 · · · vp+α | vp+α+1 · · · vn ,

which is obtained from Lk by moving vp+j to the position just before vp = s, and let zj be the extreme base associated with L•j . Proposition 10.17. For some δ ≥ 0, yk + δ(χt − χs ) can be represented as a convex combination of {zj | j = 1, . . . , α}. Proof. Put Vh = Lk (vh ) = {v1 , . . . , vh } for h = 1, . . . , n and V0 = ∅. By (10.12) we have yk (vh ) = ρ(Vh ) − ρ(Vh−1 ) (1 ≤ h ≤ n), ⎧ ρ(Vh ) − ρ(Vh−1 ) ⎪ ⎪ ⎨ ρ(Vh ∪ {vp+j }) − ρ(Vh−1 ∪ {vp+j }) zj (vh ) = ⎪ ρ(Vp−1 ∪ {vp+j }) − ρ(Vp−1 ) ⎪ ⎩ ρ(Vh ) − ρ(Vh−1 )

(1 ≤ h ≤ p − 1), (p ≤ h ≤ p + j − 1), (h = p + j), (p + j + 1 ≤ h ≤ n)

and, therefore, ⎧ =0 ⎪ ⎪ ⎨ ≤0 (zj − yk )(vh ) ≥ 0 ⎪ ⎪ ⎩ =0

(1 ≤ h ≤ p − 1), (p ≤ h ≤ p + j − 1), (h = p + j), (p + j + 1 ≤ h ≤ n),

(10.18)

where the inequalities follow from submodularity; for instance, for h = p+j, we have (zj − yk )(vp+j ) = [ρ(Vp−1 ∪ {vp+j }) − ρ(Vp−1 )] − [ρ(Vp+j ) − ρ(Vp+j−1 )], in which (Vp−1 ∪ {vp+j }) ∪ Vp+j−1 = Vp+j and (Vp−1 ∪ {vp+j }) ∩ Vp+j−1 = Vp−1 . The sign pattern of (10.18), as well as that of (χt − χs )(vh ), for h with p ≤ h ≤ p + j looks like: z1 − yk z2 − yk ··· zα − yk χt − χs

p * * * * −1

p + 1 ··· ··· p + α ⊕ 0 0 0 * ⊕ 0 0 * * ⊕ 0 * * * ⊕ 0 0 0 1

⊕ = nonnegative * = nonpositive

Note also that each row sum is equal to zero since zj (V ) = yk (V ). If all the diagonal entries marked by ⊕ are strictly positive, we can represent χt − χs as a nonnegative combination of {zj − yk | j = 1, . . . , α} with a positive coefficient for j = α; namely,

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y ) with μj ≥ 0 (j = 1, . . . , α − 1) and μα > 0. Then the χt − χs = α j=1 μj (zj − αk claim is true for δ = 1/( j=1 μj ). If a diagonal entry, say, in the j0 th row, vanishes, then zj0 = yk and the claim is true for δ = 0. Define xˆ = x + λk δ(χt − χs ). This vector can be represented as a convex combination of Yˆ = {yi | i ∈ I \ {k}} ∪ {zj | j = 1, . . . , α} by Proposition 10.17 and (10.13). Let x be the point on the line segment connecting x and x ˆ that is closest to x ˆ with the t-component x (t) ≤ 0. This means x ˆ(t) < 0 ⇒ x = x ˆ = x + λk δ(χt − χs ), x ˆ(t) ≥ 0 ⇒ x = x − x(t)(χt − χs ),

x (t) < 0, x (t) = 0.

(10.19)

Note that x can be represented as a convex combination of Yˆ ∪{yk } and, moreover, {yk } can be dispensed with if x (t) < 0. By a variant of Gaussian elimination we can obtain a convex combination representation of x using at most n vectors from Yˆ ∪ {yk }. We update {(Li , λi ) | i ∈ I} according to this representation. Since |Yˆ | + 1 ≤ 2n, step S4 can be done with O(n3 ) arithmetic operations. Proposition 10.18. The number of iterations in Schrijver’s algorithm is bounded by O(n6 ). Hence, Schrijver’s algorithm finds a minimizer of a submodular set function ρ : 2V → R with O(n8 ) function evaluations and O(n9 ) arithmetic operations, where n = |V |. Proof. Denote by β the number of indices i ∈ I such that |(s, t]≺i | = α. Let x , d , A , P  , N  , t , s , α , β  be the objects x, d, A, P, N, t, s, α, β in the next iteration. We first observe that a new arc appears only if it connects two vertices lying between s and t with respect to ≺k : (a) For each arc (v, w) ∈ A \ A we have s (k w ≺k v (k t. Proof of (a): By (v, w) ∈ / A we have w ≺k v and by (v, w) ∈ A we have v w for some j, where 1 ≤ j ≤ α and v ≺•j w means that v precedes w in L•j . Hence v = vp+j and (a) follows. The crucial properties are the monotonicity in the sense that ≺•j

(b) d (v) ≥ d(v) for all v ∈ V and, (c) if d (v) = d(v) for all v ∈ V , then (d (t ), t , s , α , β  ) is lexicographically smaller than (d(t), t, s, α, β). Proof of (b): Note that P  ⊆ P . If (b) fails, there exists an arc (v, w) ∈ A \ A with d(w) ≥ d(v)+2. By (a) we have s (k w ≺k v (k t and hence d(w) ≤ d(s)+1 = d(t) ≤ d(v) + 1, a contradiction. This shows (b). Proof of (c): Assume d (v) = d(v) for all v ∈ V . Since x (t ) < 0, we have  x(t ) < 0 or t = s by (10.19). By the choice of t and the inequality d(s) < d(t), we see that d(t ) ≤ d(t) and that if d(t ) = d(t) then t (0 t. Next assume also that t = t. We have (s , t) ∈ A , whereas (s , t) ∈ / A \ A by (a). Hence (s , t) ∈ A and the maximality of s implies s (0 s. Finally assume also that s = s. For each

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j = 1, . . . , α, (s, t]≺•j is a proper subset of (s, t]≺k . This implies α ≤ α. If α = α, then β  < β, since x (t) = x (t ) < 0 and Lk disappears in the update. Hence (c). It follows from (c) that d(v) increases for some v ∈ V in O(n4 ) iterations because each of d(t), t, s, α, β is bounded by n and there are at most n pairs (d(t), t) if d does not change. For each v ∈ V , d(v) can increase at most n times. Therefore, the total number of iterations is bounded by O(n6 ).

Note 10.19. A more detailed analysis of Vygen [208] yields an improved bound of O(n5 ) on the number of iterations in Schrijver’s algorithm.

10.2.3

Iwata–Fleischer–Fujishige’s Algorithm

We explain here Iwata–Fleischer–Fujishige’s strongly polynomial algorithm for submodular function minimization. While sharing the basic framework of section 10.2.1, this algorithm differs substantially from Schrijver’s in that it is based on a scaling technique rather than distance labeling. Weakly Polynomial Scaling Algorithm We start with a scaling algorithm for minimizing a submodular set function ρ : 2V → R. The algorithm is weakly polynomial for integer-valued ρ. It is emphasized that the value of M in (10.8) need not be computed. Recall from Proposition 10.8 that the problem dual to minimizing ρ is to maximize x− (V ) over x ∈ B(ρ). To add flexibility in solving this maximization problem we introduce a scaling parameter δ > 0 to relax (or enlarge) the feasible region B(ρ) to B(ρ + κδ ), where κδ (X) = δ · |X| · |V \ X|

(X ⊆ V ).

(10.20)

The function κδ is submodular and therefore B(ρ+ κδ ) = B(ρ)+ B(κδ ) by Theorem 4.23 (1). For a concrete representation of B(κδ ) we observe that κδ is the cut capacity function associated with a complete directed graph G = (V, A), where A = {(u, v) | u, v ∈ V, u = v} and the arc capacities are all equal to δ; indeed, κδ coincides with κ in (9.16) with c(a) = δ and c(a) = 0 for every arc a ∈ A. By a δ-feasible flow we mean a function ϕ : A → R such that for each (u, v) ∈ A we have (i) 0 ≤ ϕ(u, v) ≤ δ and (ii) either ϕ(u, v) = 0 or ϕ(v, u) = 0. Then B(κδ ) = {∂ϕ | ϕ is δ-feasible} by (9.18) in Theorem 9.3. Thus our relaxation problem with parameter δ reads as follows: Maximize z − (V ) over z = x + ∂ϕ with x ∈ B(ρ) and δ-feasible ϕ.

(10.21)

The algorithm consists of scaling phases, each of which corresponds to a fixed parameter value δ. We start with an arbitrary linear ordering, the extreme base x associated with it, zero flow ϕ = 0, and δ = min{x+ (V ), |x− (V )|}/n2 ,

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where x+ is the vector in RV defined by x+ (v) = max(0, x(v)) for v ∈ V . In each scaling phase, we construct an approximate solution to (10.21) from a given pair of a base x and a δ-feasible flow ϕ and then cut δ and ϕ in half for the next scaling phase. We terminate the algorithm when δ is sufficiently small; specifically, when δ < 1/n2 for integer-valued ρ. In the scaling phase with parameter value δ we maintain a δ-feasible flow ϕ and a directed graph Gϕ = (V, Aϕ ) with arc set Aϕ = {(u, v) | u, v ∈ V, u = v, ϕ(u, v) = 0}.

(10.22)

We aim at increasing z − (V ) by sending flow along directed paths in Gϕ from S to T defined by S = {v ∈ V | z(v) ≤ −δ},

T = {v ∈ V | z(v) ≥ +δ}.

Such a directed path is called a δ-augmenting path. We also maintain a base x represented in the form of (10.13) with {(Li , λi ) | i ∈ I}. If there exists a δ-augmenting path P , we modify ϕ to another δ-feasible flow by setting ϕ(u, v) := δ − ϕ(v, u), ϕ(v, u) := 0 for each arc (u, v) in P . This results in an increase of z − (V ) by δ, an improvement in the objective function in our optimization problem (10.21). We refer to this operation as Augment(ϕ, P ). Suppose that no δ-augmenting path exists and denote by W the set of vertices reachable from S in Gϕ ; we have S ⊆ W ⊆ V \ T . In this case we cannot increase z − (V ) by flow augmentation but the current solution may or may not be optimal. With an additional condition we have approximate optimality for (10.21), as is stated in the following theorem, which is a relaxation version of the min-max relation in Proposition 10.8. A triple (i, u, v) of i ∈ I, u ∈ W , and v ∈ V \ W is called active if v is the immediate predecessor of u in Li . Proposition 10.20. If S ⊆ W ⊆ V \ T , no arcs leave W in Gϕ , and no active triples exist, then z − (V ) ≥ ρ(W ) − nδ,

x− (V ) ≥ ρ(W ) − n2 δ.

(10.23)

Moreover, W is a minimizer of ρ if δ < Δ/n2 with Δ ≤ min{ρ(X) − ρ(Y ) | X, Y ⊆ V, ρ(X) − ρ(Y ) > 0}.

(10.24)

Proof. Since S ⊆ W ⊆ V \ T , we have z(v) < δ for every v ∈ W and z(v) > −δ for every v ∈ V \ W . Therefore, z − (V ) = z − (W ) + z − (V \ W ) ≥ z(W ) − δ|W | − δ|V \ W | = x(W ) + ∂ϕ(W ) − nδ. Since x(W ) = ρ(W ) by the nonexistence of active triples and Proposition 10.9 (2) and ∂ϕ(W ) ≥ 0 by the nonexistence of arcs leaving W in Gϕ , we have z − (V ) ≥ ρ(W ) − nδ. Since ∂ϕ(v) ≤ (n − 1)δ for every v ∈ V , we have x− (V ) ≥ z − (V ) − n(n − 1)δ ≥ ρ(W ) − n2 δ.

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With δ < Δ/n2 we have x− (V ) ≥ ρ(W ) − n2 δ > ρ(W ) − Δ, whereas x− (V ) ≤ ρ(Y ) for all Y ⊆ V by (10.10). Hence W is a minimizer of ρ. On the basis of Proposition 10.20 above we terminate the scaling phase if neither augmenting path nor active triple exists. Otherwise, while keeping z invariant, we aim at “improving the situation” by either 1. eliminating an active triple or 2. enlarging the reachable set W . With a view to eliminating an active triple (i, u, v) we modify Li by swapping u and v in Li ; denote the old pair (Li , yi ) by (Lk , yk ) with a new index k. The extreme base associated with the updated Li is given by yi = yk + β(χu − χv ) with β = ρ(Lk (u) \ {v}) − ρ(Lk (u)) + yk (v) (see (10.12)). Defining α = min(ϕ(u, v), λi β) let us modify x and ϕ by setting x := x + α(χu − χv ),

ϕ(u, v) := ϕ(u, v) − α.

Then z = x + ∂ϕ is invariant and ϕ remains δ-feasible. The updated x is equal to    α α λi − λj yj , yk + yi + β β

(10.25)

j∈I\{i}

which is a convex combination of {yj | j ∈ I ∪ {k}}. In the saturating case where α = λi β, the old extreme base yk disappears from (10.25). Since u precedes v in the new Li , the active triple (i, u, v) is successfully eliminated, whereas the size of the index set I remains the same. In the nonsaturating case where α < λi β, the old extreme base yk remains and the size of I increases by one. Nevertheless, the situation is somewhat improved; namely, the reachable set W is enlarged to contain v as a result of ϕ(u, v) = 0 for the updated flow ϕ. The above task is done by the procedure Double-Exchange(i, u, v) below. Procedure Double-Exchange(i, u, v) S1: Set β := ρ(Li (u) \ {v}) − ρ(Li (u)) + yi (v), α := min(ϕ(u, v), λi β). S2: If α < λi β, then let k be a new index and set I := I ∪ {k}, λk := λi − α/β, λi := α/β, yk := yi , Lk := Li . S3: Set yi := yi + β(χu − χv ), x := x + α(χu − χv ), ϕ(u, v) := ϕ(u, v) − α. Update Li by swapping u and v. The overall structure of the algorithm, which we name the IFF scaling algorithm, is described below. Reduce(x, I) is a procedure that computes an expression of x as a convex combination of at most n affinely independent extreme bases chosen from the current extreme bases indexed by I; this can be done by a variant of Gaussian elimination. Parameter Δ for the stopping criterion in step S4 should satisfy (10.24); we take Δ = 1 for integer-valued ρ.

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IFF scaling algorithm for submodular function minimization S0: Take any linear ordering L1 and let y1 be the associated extreme base. If y1+ (V ) = 0, then output V as a minimizer and stop. If y1− (V ) = 0, then output ∅ as a minimizer and stop. Set I := {1}, λ1 := 1, x := y1 , ϕ := 0, δ := min{x+ (V ), |x− (V )|}/n2 . S1: Let W be the set of vertices reachable from S in Gϕ . S2: If W ∩ T = ∅, then let P be a δ-augmenting path, apply Augment(ϕ, P ) and Reduce(x, I), and go to S1. S3: If there exists an active triple, then apply Double-Exchange to an active triple (i, u, v) and go to S1. S4: If δ < Δ/n2 , then output W as a minimizer and stop. S5: Apply Reduce(x, I), set δ := δ/2, and ϕ := ϕ/2, and go to S1. Iterations of steps S1 to S3 constitute a scaling phase. Step S2 increases z − (V ) by flow augmentation, whereas step S3 improves the situation by Double-Exchange. Whenever flow is augmented in step S2 we apply Reduce to reduce the size of I, which may have grown as a result of repeated executions of step S3. The correctness of the algorithm follows from the second half of Proposition 10.20, which guarantees the optimality of W at the termination in step S4. Note, however, that the base x is not necessarily optimal in (10.11). As for complexity the algorithm is weakly polynomial for integer-valued ρ. Proposition 10.21. The IFF scaling algorithm finds a minimizer of an integervalued submodular set function ρ : 2V → Z with O(n5 log2 M ) function evaluations and arithmetic operations, where n = |V | and M = max{|ρ(X)| | X ⊆ V }. Proof. This can be derived from the properties listed in Proposition 10.22. Proposition 10.22. (1) The number of scaling phases is O(log2 (M/Δ)). (2) The first scaling phase calls Augment O(n2 ) times. (3) A subsequent scaling phase calls Augment O(n2 ) times. (4) Between calls to Augment, there are at most n − 1 calls to nonsaturating Double-Exchange. (5) Between calls to Augment, there are at most 2n3 calls to saturating DoubleExchange. (6) We always have |I| < 2n. (7) Reduce(x, I) with |I| < 2n can be done in O(n3 ) arithmetic operations. Proof. (1) We have δ ≤ M/n2 in step S0, since x+ (V ) = x(X) ≤ ρ(X) ≤ M for X = supp+ (x). The number of scaling phases is bounded by log2 ((M/n2 )/(Δ/n2 )) = log2 (M/Δ). (2) Let x denote the initial base in step S0. Then z − (V ) = x− (V ) at the beginning of the scaling phase. Throughout the scaling phase we have z − (V ) ≤ z(V ) = x(V ) as well as z − (V ) ≤ 0. Since Augment increases z − (V ) by δ, the number of calls to Augment is bounded by

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(3) At the beginning of a subsequent scaling phase, we have zˆ− (V ) ≥ ρ(W )−nδˆ by Proposition 10.20, where zˆ = x + ∂ ϕˆ with ϕˆ = 2ϕ and δˆ = 2δ. Since z (X) | X ⊆ V } = min{z(X) + ∂ϕ(X) | X ⊆ V } zˆ− (V ) = min{ˆ ≤ min{z(X) + n2 δ/4 | X ⊆ V } = z − (V ) + n2 δ/4, this implies that z − (V ) ≥ ρ(W )− 2nδ − n2δ/4 at the beginning of the scaling phase. Throughout the scaling phase we have z − (V ) ≤ z(W ) = x(W ) + ∂ϕ(W ) ≤ ρ(W ) + n2 δ/4. Therefore, the number of calls to Augment is bounded by (2nδ + n2 δ/2)/δ = 2n + n2 /2. (4) Each nonsaturating Double-Exchange adds a new element to W . (5), (6) A call to Reduce results in |I| ≤ n. A new index is added to I only in a nonsaturating Double-Exchange. By (4), |I| grows to at most 2n − 1. Hence, the number of triples (i, u, v) is bounded by 2n3 . (7) Reduce can be performed by a variant of Gaussian elimination. The following is a key property of the scaling algorithm that we make use of in designing a strongly polynomial algorithm. Recall that a scaling phase ends when the algorithm reaches step S4. Proposition 10.23. At the end of a scaling phase with parameter δ, the following hold true. (1) If x(w) < −n2 δ, then w is contained in every minimizer of ρ. (2) If x(w) > n2 δ, then w is not contained in any minimizer of ρ. Proof. We have x− (V ) ≥ ρ(W ) − n2 δ by Proposition 10.20, whereas, for any minimizer X of ρ, we have ρ(W ) ≥ ρ(X) ≥ x(X) ≥ x− (X). Hence, x− (V ) ≥ x− (X) − n2 δ. Therefore, if x(w) < −n2 δ, then w ∈ X. On the other hand, x− (X) ≥ x− (V ) ≥ ρ(W ) − n2 δ ≥ x(X) − n2 δ. / X. Therefore, if x(w) > n2 δ, then w ∈

Strongly Polynomial Fixing Algorithm Using the scaling algorithm as a subroutine, we can devise a strongly polynomial algorithm for submodular function minimization. The strongly polynomial algorithm, which we call the IFF fixing algorithm, exploits two fundamental facts: • The minimizers of ρ form a ring family (see Note 4.8).

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• A ring family can be represented as the set of ideals of a directed graph (see Note 4.7). Let D◦ be the directed graph representing the family D◦ of minimizers of ρ. This means that W ⊆ V is a minimizer of ρ if and only if it is an ideal of D◦ such that min D◦ ⊆ W ⊆ max D◦ . The algorithm aims at constructing the graph by identifying arcs of D◦ or elements of (min D◦ ) ∪ (V \ max D◦ ) one by one with the aid of the scaling algorithm. To be more specific, we maintain an acyclic graph D = (U, F ) and two disjoint subsets Z and H of V such that • Z is included in every minimizer of ρ, i.e., Z ⊆ min D◦ ; • H is disjoint from any minimizer of ρ, i.e., H ⊆ V \ max D◦ ; • each u ∈ U corresponds to a nonempty subset, say, Γ(u), of V , and {Γ(u) | u ∈ U } is a partition of V \ (Z ∪ H); • for each u ∈ U and any minimizer W of ρ, either Γ(u) ⊆ W or Γ(u) ∩ W = ∅; • an arc (u, w) ∈ F implies that every minimizer of ρ including Γ(u) includes Γ(w). 5 Using the notation Γ(Y ) = u∈Y Γ(u) for Y ⊆ U , we define a function ρ˜ : 2U → R by ρ˜(Y ) = ρ(Γ(Y ) ∪ Z) − ρ(Z) (Y ⊆ U ). It is easy to verify that • ρ˜ is submodular; • a subset W of V is a minimizer of ρ if and only if W = Γ(Y ) ∪ Z for a minimizer Y ⊆ U of ρ˜; • an arc (u, w) ∈ F implies that every minimizer of ρ˜ containing u contains w; i.e., a minimizer of ρ˜ is an ideal of D. The algorithm consists of iterations. Initially we set U := V , Z := ∅, H := ∅, and F := ∅. At the beginning of each iteration, we compute η = max{ρ˜(R(u)) − ρ˜(R(u) \ {u}) | u ∈ U },

(10.26)

where R(u) denotes the set of vertices reachable from u ∈ U in D. If η ≤ 0, we are done by Proposition 10.24 below. Otherwise, we either enlarge Z ∪ H or add an arc to D, where directed cycles that may possibly arise in this modification are contracted to a single vertex; the partition Γ of V \ (Z ∪ H) is modified accordingly. Proposition 10.24. If η ≤ 0, then V \ H is the maximal minimizer of ρ. Proof. Let Y be the unique maximal minimizer of ρ˜. If Y = U , there is an element u ∈ U \ Y such that Y ∪ {u} is an ideal of D. By Y ∪ {u} ⊇ R(u) and the submodularity of ρ˜, we have ρ˜(Y ∪ {u}) − ρ˜(Y ) ≤ ρ˜(R(u)) − ρ˜(R(u) \ {u}) ≤ 0,

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which contradicts the definition of Y . Thus, U is the maximal minimizer of ρ˜ and hence Γ(U ) ∪ Z = V \ H is the maximal minimizer of ρ. Suppose that η > 0 and let uˆ ∈ U be the vertex that attains the maximum in (10.26). Then η = ρ˜(U ) − ρ˜(R(ˆ u) \ {ˆ u}) + [˜ ρ(R(ˆ u)) − ρ˜(U )] and we have at least one of the following three cases: (i) ρ˜(U ) ≥ η/3, (ii) ρ˜(R(ˆ u) \ {ˆ u}) ≤ −η/3, or (iii) ρ˜(R(ˆ u)) − ρ˜(U ) ≥ η/3. ρ, D, η), described below, Case (i): If ρ˜(U ) ≥ η/3, we invoke a procedure Fix+ (˜ to find an element w ∈ U that is not contained in any minimizer of ρ˜. Since Γ(w) cannot be included in any minimizer of ρ, we add Γ(w) to H and delete w from D. ρ, D, η), Case (ii): If ρ˜(R(ˆ u) \ {ˆ u}) ≤ −η/3, we invoke another procedure Fix− (˜ described below, to find an element w ∈ U that is contained in every minimizer of ρ˜. Since Γ(w) must be included in every minimizer of ρ, we add Γ(w) to Z and delete w from D. Case (iii): If ρ(R(ˆ ˜ u)) − ρ˜(U ) ≥ η/3, we consider the contraction of ρ˜ by R(ˆ u), u) defined by which is a submodular function ρ˜∗ on U \ R(ˆ u)) − ρ˜(R(ˆ u)) ρ˜∗ (X) = ρ˜(X ∪ R(ˆ

(X ⊆ U \ R(ˆ u)),

and find an element w ∈ U \ R(ˆ u) that is contained in every minimizer of ρ˜∗ . As explained below, we can do this by applying Fix− to (˜ ρ∗ , D∗ , η), where D∗ means the subgraph of D induced on the vertex set U \ R(ˆ u). A subset X ⊆ U \ R(ˆ u) is u) minimizes ρ˜ over subsets of U containing a minimizer of ρ˜∗ if and only if X ∪ R(ˆ R(ˆ u). Therefore, if a minimizer of ρ˜ containing u ˆ exists, then it must contain w. Equivalently, if a minimizer of ρ including Γ(ˆ u) exists, then it must include Γ(w). Accordingly, we add a new arc (ˆ u, w) to F , where the arc (ˆ u, w) is new because w∈ / R(ˆ u). If the added arc yields directed cycles, we contract the cycles to a single vertex, with corresponding modifications of U and Γ. Thus, in each iteration with η > 0, we either enlarge Z ∪ H or add a new arc to D. Therefore, after at most n2 iterations, we can terminate the algorithm with η ≤ 0 when we have a minimizer of ρ by Proposition 10.24. ρ, D, η) and Fix+ (˜ ρ, D, η) are as follows. Given a subThe procedures Fix− (˜ U modular function ρ˜ : 2 → R, an acyclic graph D = (U, F ), and a positive real number η such that ρ˜(Y ) ≤ −η/3 for some Y ⊆ U ,

(10.27)

the procedure Fix− (˜ ρ, D, η) finds an element w ∈ U that is contained in every minimizer of ρ˜. Similarly, Fix+ (˜ ρ, D, η) finds an element w ∈ U that is not contained in any minimizer of ρ˜ when ρ˜(U ) ≥ η/3.

(10.28)

The procedures Fix− (˜ ρ, D, η) and Fix+ (˜ ρ, D, η) are the same as the IFF scaling algorithm except that they start with δ = η and a linear extension of the partial order represented by D and return w in step S4. We put n ˜ = |U |.

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˜ D, η) Procedure Fix− (ρ, S0: Take any linear extension L1 of the partial order represented by D and let y1 be the associated extreme base. Set I := {1}, λ1 := 1, x := y1 , ϕ := 0, δ := η. S1: Let W be the set of vertices reachable from S in Gϕ . S2: If W ∩ T = ∅, then let P be a δ-augmenting path, apply Augment(ϕ, P ) and Reduce(x, I), and go to S1. S3: If there exists an active triple (i, u, v), then apply Double-Exchange to it and go to S1. S4: If there exists w ∈ U with x(w) < −˜ n2 δ, then return such a w. S5: Apply Reduce(x, I), set δ := δ/2, and ϕ := ϕ/2, and go to S1. ρ, D, η) is identical to Fix− (˜ ρ, D, η) except that step S4 is replaced Procedure Fix+ (˜ with S4: If there exists w ∈ U with x(w) > n ˜ 2 δ, then return such a w. The correctness of the above procedures at the termination in step S4 is guaranteed by Proposition 10.23. As to the complexity we have the following as well as Proposition 10.22 (3)–(7). ρ, D, η). Proposition 10.25. The following statements hold true for Fix± (˜ (1) The number of scaling phases is O(log2 n ˜ ), where n ˜ = |U |. (2) If y(u) ≤ η for any u ∈ U and any extreme base y for ρ˜ generated by a linear extension of the partial order of D,

(10.29)

then the first scaling phase calls Augment O(˜ n) times. ˜3δ Proof. (1) Assume δ < η/(3˜ n3 ). By (10.28) we have x(U ) = ρ˜(U ) ≥ η/3 > n 2 and hence x(w) > n ˜ δ for some w ∈ U . Therefore, the number of scaling phases in Fix+ is bounded by log2 (3˜ n3 ) = O(log2 n ˜ ). For Y in (10.27), we have x(Y ) ≤ 3 ρ˜(Y ) ≤ −η/3 < −˜ n δ and hence x(w) < −˜ n2 δ for some w ∈ Y . Therefore, the − ˜ ). number of scaling phases in Fix is bounded by O(log2 n (2) Let x denote the initial base in step S0. By the proof of Proposition 10.22 (2), the number of calls to Augment is bounded by x+ (U )/δ, whereas x+ (U ) ≤ n ˜ η by (10.29). Since δ = η, the number of calls to Augment is bounded by n ˜. The applications of Fix± in the IFF fixing algorithm are legitimate. Proposition 10.26. Let η be defined by (10.26). (1) Conditions (10.28) and (10.29) are satisfied by (˜ ρ, D, η) in Case (i). (2) Conditions (10.27) and (10.29) are satisfied by (˜ ρ, D, η) in Case (ii). (3) Conditions (10.27) and (10.29) are satisfied by (˜ ρ∗ , D∗ , η) in Case (iii). Proof. In Case (i), (10.28) is obviously satisfied. Condition (10.27) is satisfied with Y = R(ˆ u) \ {ˆ u} in Case (ii) and Y = U \ R(ˆ u) in Case (iii). To show (10.29) for

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(˜ ρ, D, η), let y be an extreme base generated by a linear extension of the partial order of D. For each u ∈ U we have y(u) = ρ˜(Y ) − ρ˜(Y \ {u}) for some Y ⊇ R(u), whereas ρ˜(Y ) − ρ˜(Y \ {u}) ≤ ρ˜(R(u)) − ρ˜(R(u) \ {u}) ≤ η by the submodularity of ρ˜ and the definition of η. This proves (10.29) for (˜ ρ, D, η). u) and observe Finally, for each u ∈ U \ R(ˆ u), put R∗ (u) = R(u) \ R(ˆ ρ˜∗ (R∗ (u)) − ρ˜∗ (R∗ (u) \ {u}) = ρ˜(R(u) ∪ R(ˆ u)) − ρ˜((R(u) \ {u}) ∪ R(ˆ u)) ≤ ρ˜(R(u)) − ρ˜(R(u) \ {u}) ≤ η. This shows (10.29) for (˜ ρ∗ , D∗ , η). We are now in the position to assert the correctness and the strong polynomiality of the IFF fixing algorithm. Proposition 10.27. The IFF fixing algorithm finds a minimizer of a submodular set function ρ : 2V → R with O(n7 log2 n) function evaluations and arithmetic operations, where n = |V |. Proof. This follows from Proposition 10.25 and Proposition 10.22 (3)–(7). Finally, we note that the maximal minimizer is found. Proposition 10.28. The IFF fixing algorithm finds the maximal minimizer of ρ. Proof. This follows from Proposition 10.24. Note 10.29. For minimization of a submodular set function ρ : 2V → R ∪ {+∞} defined effectively on a general ring family, the IFF scaling/fixing algorithm can be applied to the associated finite-valued submodular function ρ in Note 10.14. Alternatively, the IFF scaling/fixing algorithm can be tailored to this general case if the ring family is represented as the set of ideals of a directed graph (V, E). The min-max relation (10.11) in Proposition 10.8 holds true in this general case. The representation (10.13) of a base x as a convex combination of extreme bases yi (i ∈ I) should be augmented by an additional term ∂ξ as 8 9   λi yi + ∂ξ with λi = 1, λi > 0 (i ∈ I) , (10.30) x= i∈I

i∈I

where ∂ξ is the boundary of a nonnegative flow ξ : E → R+ . Then we have z = x + ∂ϕ =

 i∈I

λi yi + ∂ξ + ∂ϕ =

 i∈I

λi yi + ∂ψ,

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where ψ is a flow in the complete graph on V representing the superposition of ξ and ϕ. It is possible to design an algorithm that finds the minimum of ρ by maintaining the extreme bases and the flow ψ. See Iwata [99] for more details.

10.3

Minimization of L-Convex Functions

Three kinds of algorithms for L-convex function minimization are described: the steepest descent algorithm, the steepest descent scaling algorithm, and the reduction to submodular function minimization on a distributive lattice. All of them depend heavily on the algorithms for submodular function minimization in section 10.2. Throughout this section g : ZV → R ∪ {+∞} denotes an L- or L -convex function with |V | = n. For an L-convex function g, it is assumed that g(p + 1) = g(p)

(∀ p ∈ ZV ),

(10.31)

since otherwise g does not have a minimum.

10.3.1

Steepest Descent Algorithm

The local characterization of global minimality for L-convex functions (Theorem 7.14) naturally leads to the following steepest descent algorithm. Steepest descent algorithm for an L-convex function g ∈ L[Z → R] S0: Find a vector p ∈ dom g. S1: Find X ⊆ V that minimizes g(p + χX ). S2: If g(p) ≤ g(p + χX ), then stop (p is a minimizer of g). S3: Set p := p + χX and go to S1. Step S1 amounts to minimizing a set function ρp (X) = g(p + χX ) − g(p)

(10.32)

over all subsets X of V . As a consequence of the submodularity of g, ρp is submodular and can be minimized in strongly polynomial time by the algorithms in section 10.2. At the termination in step S2, p is a global minimizer by Theorem 7.14 (1) (L-optimality criterion). The function value g decreases monotonically with iterations. This property alone does not ensure finite termination in general, but it does if g is integer valued and bounded from below. We introduce a tie-breaking rule in step S1: take the (unique) minimal minimizer X of ρp .

(10.33)

Thus, we can guarantee an upper bound on the number of iterations. Let p◦ be the initial vector found in step S0. If g has a minimizer at all, it has a minimizer p∗ satisfying p◦ ≤ p∗ by (10.31). Let p∗ denote the smallest of such minimizers, which exists since p∗ ∧ q ∗ ∈ arg min g for p∗ , q ∗ ∈ arg min g.

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Proposition 10.30. In step S1, p ≤ p∗ implies p + χX ≤ p∗ . Hence the number of iterations is bounded by ||p◦ − p∗ ||1 . Proof. Put Y = {v ∈ V | p(v) = p∗ (v)} and p = p + χX . By submodularity we have g(p∗ ) + g(p ) ≥ g(p∗ ∨ p ) + g(p∗ ∧ p ), whereas g(p∗ ) ≤ g(p∗ ∨p ) since p∗ is a minimizer of g. Hence g(p ) ≥ g(p∗ ∧p ). Here we have p = p + χX and p∗ ∧ p = p + χX\Y , whereas X is the minimal minimizer by the tie-breaking rule (10.33). This means that X \ Y = X; i.e., X ∩ Y = ∅. Therefore, p = p + χX ≤ p∗ . It is easy to find the minimal minimizer of ρp using the existing algorithms for submodular set function minimization (see Notes 10.11, 10.12, and 10.13 and Proposition 10.28). Assuming that the minimal minimizer of a submodular set function can be computed with O(σ(n)) function evaluations and O(τ (n)) arithmetic operations and denoting by F an upper bound on the time to evaluate g, we can perform step S1 in O(σ(n)F + τ (n)) time. We measure the size of the effective domain of g by ˆ 1 = max{||p − q||1 | p, q ∈ dom g, p(v) = q(v) for some v ∈ V }, K

(10.34)

where it is noted that dom g itself is unbounded by (10.31). ˆ 1 , the number of Proposition 10.31. For an L-convex function g with finite K iterations in the steepest descent algorithm with tie-breaking rule (10.33) is bounded ˆ 1 . Hence, if a vector in dom g is given, the algorithm finds a minimizer of g by K ˆ 1 ) time. in O((σ(n)F + τ (n))K ˆ 1 since p◦ (v) = p∗ (v) for some v ∈ V . Then the Proof. We have ||p◦ − p∗ ||1 ≤ K claim follows from Proposition 10.30. The steepest descent algorithm can be adapted to L -convex functions. Let g be an L -convex function and recall from (7.2) that it is associated with an L-convex function g˜ as (p0 ∈ Z, p ∈ ZV ). (10.35) g˜(p0 , p) = g(p − p0 1) The steepest descent algorithm above applied to this L-convex function g˜ yields the following algorithm for the L -convex function g. Steepest descent algorithm for an L -convex function g ∈ L [Z → R] S0: Find a vector p ∈ dom g. S1: Find ε ∈ {1, −1} and X ⊆ V that minimize g(p + εχX ). S2: If g(p) ≤ g(p + εχX ), then stop (p is a minimizer of g). S3: Set p := p + εχX and go to S1. Step S1 amounts to minimizing a pair of submodular set functions ρ+ p (X) = g(p + χX ) − g(p),

ρ− p (X) = g(p − χX ) − g(p).

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− be the maximal minimizer of ρ− Let X + be the minimal minimizer of ρ+ p and X p. The tie-breaking rule in step S1 above reads  − (1, X + ) if min ρ+ p ≤ min ρp , (ε, X) = (10.36) − + (−1, X ) if min ρp > min ρ− p.

This is a translation of the tie-breaking rule (10.33) for g˜ in (10.35) through the correspondence g p → p + χX p → p − χX

g˜ ⇐⇒ p˜ → p˜ + (0, χX ) ⇐⇒ p˜ → p˜ + (1, χV \X )

where p˜ = (0, p) ∈ Z × ZV . Since (1, χV \X − ) cannot be minimal in the presence of − (0, χX + ), we choose (1, X + ) in the case of min ρ+ p = min ρp . At the termination in step S2, p is a global minimizer by Theorem 7.14 (2) (L-optimality criterion). In view of the complexity bound given in Proposition 10.31 we will derive a ˆ 1 (˜ g ) be defined by bound on the size of dom g˜ in terms of the size of dom g. Let K (10.34) for g˜. The 1 -size and ∞ -size of dom g are denoted, respectively, by K1 = max{||p − q||1 | p, q ∈ dom g},

(10.37)

K∞ = max{||p − q||∞ | p, q ∈ dom g}.

(10.38)

ˆ 1 (˜ g ) ≤ K1 + nK∞ ≤ min[(n + 1)K1 , 2nK∞ ]. Proposition 10.32. K ˆ 1 (˜ g ) = |p0 − q0 | + Proof. Take p˜ = (p0 , p) and q˜ = (q0 , q) in dom g˜ such that K ||p − q||1 and either (i) p0 = q0 or (ii) p(v) = q(v) for some v ∈ V . We may assume p0 ≥ q0 and p ≥ q since p˜ ∨ q˜, p˜ ∧ q˜ ∈ dom g˜ and ||(˜ p ∨ q˜) − (˜ p ∧ q˜)||1 = ||˜ p − q˜||1 . The vectors p = p − p0 1 and q  = q − q0 1 belong to dom g. In case (i), we have ˆ 1 (˜ K g ) = ||p− q||1 = ||p − q  ||1 ≤ K1 . In case (ii), we have p0 − q0 = q  (v)− p (v) and ˆ 1 (˜ K g ) = |p0 − q0 | + ||p − q||1  = (p0 − q0 ) + (p(u) − q(u)) u∈V



= (p0 − q0 ) + =



(p (u) − q  (u)) + n(p0 − q0 )

u∈V 

(p (u) − q  (u)) − n(p (v) − q  (v))

u =v

≤ K1 + nK∞ . Note finally that K1 ≤ nK∞ and K∞ ≤ K1 . The steepest descent algorithm could be used for minimizing quasi L-convex functions satisfying (SSQSBw ) because of Theorem 7.53 (quasi L-optimality criterion). Note, however, that the set function ρp of (10.32), to be minimized in step S1, is not necessarily submodular and hence no efficient procedure is available for step S1.

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Chapter 10. Algorithms

Steepest Descent Scaling Algorithm

The steepest descent algorithm for L-convex function minimization can be made more efficient with the aid of a scaling technique. The efficiency of the resulting steepest descent scaling algorithm is guaranteed by the complexity analysis in section 10.3.1 combined with the proximity theorem for L-convex functions. The algorithm for an L-convex function g with (10.31) reads as follows, where ˆ ∞ = max{||p − q||∞ | p, q ∈ dom g, p(v) = q(v) for some v ∈ V }. K Steepest descent scaling algorithm for an L-convex function g ∈ L[Z → R] ˆ S0: Find a vector p ∈ dom g and set α := 2log2 (K∞ /2n) . S1: Find an integer vector q that locally minimizes g˜(q) = g(p + αq) in the sense of g˜(q) ≤ g˜(q + χX ) (∀ X ⊆ V ) by the steepest descent algorithm of section 10.3.1 with initial vector 0 and set p := p + αq. S2: If α = 1, then stop (p is a minimizer of g). S3: Set α := α/2 and go to S1. Note first that the function g˜(q) = g(p + αq) is an L-convex function. By the L-proximity theorem (Theorem 7.18 (1)), there exists a minimizer q of g˜ satisfying 0 ≤ q ≤ (n − 1)1. Then, by Propositions 10.30 and 10.31, the steepest descent algorithm with tie-breaking rule (10.33) finds the minimizer in step S1 in O((σ(n)F + τ (n))n2 ) time, where σ(n), τ (n), and F are defined in section 10.3.1. ˆ ∞ /2n) and, at the The number of executions of step S1 is bounded by log2 (K termination of the algorithm in step S2 with α = 1, p is a minimizer of g by Theorem 7.14 (L-optimality criterion). Thus, the complexity of the steepest descent ˆ ∞ /2n). scaling algorithm is bounded by a polynomial in n and log2 (K The steepest descent scaling algorithm can be adapted to quasi L-convex functions satisfying (SSQSB) because of Theorem 7.53 (quasi L-optimality criterion) and Theorem 7.54 (quasi L-proximity theorem). Note, however, that no efficient procedure is available for the minimization in step S1.

10.3.3

Reduction to Submodular Function Minimization

The effective domain of an L -convex function g is a distributive lattice (a sublattice of ZV ) on which g is submodular. Hence we can make use of submodular function minimization algorithms as adapted to functions on distributive lattices (see Note 10.15). If the 1 -size of dom g is given by K1 , dom g is isomorphic to a sublattice ˜ of the Boolean lattice 2V for a set V˜ of cardinality K1 . Hence, the complexity of this algorithm is polynomial in n and K1 . It may be noted, however, that, being dependent only on the submodularity of g, this approach does not fully exploit L -convexity.

10.4

Algorithms for M-Convex Submodular Flows

Five algorithms for the M-convex submodular flow problem are described: the twostage algorithm, the successive shortest path algorithm, the cycle-canceling algo-

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rithm, the primal-dual algorithm, and the conjugate scaling algorithm. Because the optimality criterion for the M-convex submodular flow problem is essentially equivalent to the duality theorems for M-/L-convex functions, these algorithms can be used for finding a separating affine function in the separation theorem and the optimal solutions in the minimization/maximization problems in the Fenchel-type duality.

10.4.1

Two-Stage Algorithm

This section is intended to provide a general structural view on the duality nature of the M-convex submodular flow problem. It is based on the recognition of the M-convex submodular flow problem as a composition of the Fenchel-type duality and the minimum cost flow problem that does not involve an M-convex function. The algorithm presented in this section, called the two-stage algorithm, computes an optimal potential by solving an L-convex minimization problem in the dual problem and constructs an optimal flow as a feasible flow to another submodular flow problem. As an adaptation of our discussion in section 9.1.4, the relationship between the M-convex submodular flow problem MSFP3 and the Fenchel-type duality may be summarized as follows. To be specific, we consider the integer-flow version of MSFP3 on the graph G = (V, A) with f ∈ M[Z → Z] and fa ∈ C[Z → Z] for a ∈ A. M-convex submodular flow problem MSFP3 (integer flow) Minimize

Γ3 (ξ) =



fa (ξ(a)) + f (∂ξ)

(10.39)

a∈A

subject to ξ(a) ∈ dom fa

(a ∈ A),

∂ξ ∈ dom f, ξ(a) ∈ Z (a ∈ A).

(10.40) (10.41) (10.42)

We assume the existence of an optimal solution. First, we identify the problem dual to MSFP3 and indicate how to compute an optimal potential. With the introduction of a function ) * ( )  ) fa (ξ(a))) ∂ξ = x (x ∈ ZV ) fA (x) = inf ) ξ∈ZA a∈A

we obtain inf Γ3 (ξ) = inf [f (x) + fA (x)] ,

ξ∈ZA

x∈ZV

where f ∈ M[Z → Z] and fA ∈ M[Z → Z]. Putting g = f • , ga = fa • for a ∈ A, and  ga (δp(a)) (p ∈ ZV ), gA (p) = a∈A

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we have gA = fA • (see (9.28)) and also g ∈ L[Z → Z], ga ∈ C[Z → Z] for a ∈ A, and gA ∈ L[Z → Z]. The Fenchel-type duality (Theorem 8.21 (3)) gives inf [f (x) + fA (x)] = − inf [g(p) + gA (−p)] ,

x∈ZV

p∈ZV

which is equivalent to61 , , + +   inf f (∂ξ) + fa (ξ(a)) = − inf g(p) + ga (−δp(a)) . ξ∈ZA

p∈ZV

a∈A

The function g˜(p) = g(p) +



(10.43)

(10.44)

a∈A

ga (−δp(a))

(10.45)

a∈A

to be minimized on the right-hand side of (10.44) is an L-convex function and a minimizer of g˜ is an optimal potential for MSFP3 , and vice versa, in the sense of Theorem 9.16. Next we discuss how to construct an optimal flow. Let p∗ be a minimizer of g˜ and define c∗ : A → Z ∪ {−∞}, c∗ : A → Z ∪ {+∞}, and B ∗ ⊆ ZV by [c∗ (a), c∗ (a)]Z = arg min fa [δp∗ (a)] ∗

(a ∈ A),



B = arg min f [−p ],

(10.46) (10.47)





where B is an M-convex set by Proposition 6.29. Since p is an optimal potential, a flow ξ ∗ is optimal if and only if c∗ (a) ≤ ξ ∗ (a) ≤ c∗ (a)

(a ∈ A),

∂ξ ∗ ∈ B ∗ .

(10.48)

The two-stage algorithm is described as follows. Two-stage algorithm for MSFP3 (integer flow) S1: Find a minimizer p∗ of g˜ in (10.45). S2: Find a flow ξ ∗ satisfying (10.48). The feasibility of this approach is guaranteed by the following facts if the given functions, f and fa for a ∈ A, can be evaluated. 1. We can evaluate g by applying an M-convex function minimization algorithm to f . Similarly, we can evaluate ga for a ∈ A. 2. We can find a minimizer p∗ in step S1 by applying an L-convex function minimization algorithm to g˜. 3. We can find a member of B ∗ by applying an M-convex function minimization algorithm to f [−p∗ ]. 4. We can find a flow ξ ∗ in step S2 as a feasible flow to the submodular flow problem defined by c∗ , c∗ , and B ∗ . This can be done, e.g., by the successive shortest path algorithm described in section 10.4.2. 61 We

have seen (10.44) in (9.83) as a special case of Theorem 9.26 (3).

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10.4.2

311

Successive Shortest Path Algorithm

We present the successive shortest path algorithm to find a feasible integer flow in an integral submodular flow problem. We adopt the most primitive form of the algorithm to better explain the basic idea without being bothered by technicalities. Given a graph G = (V, A), an upper capacity c : A → Z ∪ {+∞}, a lower capacity c : A → Z ∪ {−∞}, and an M-convex set B ⊆ ZV , we are to find an integer flow ξ : A → Z satisfying c(a) ≤ ξ(a) ≤ c(a) ∂ξ ∈ B.

(a ∈ A),

(10.49) (10.50)

It is assumed that c(a) ≤ c(a) for each a ∈ A. The algorithm maintains a pair (ξ, x) ∈ ZA × ZV of integer flow ξ satisfying (10.49) and base x ∈ B and repeats modifying (ξ, x) to resolve the discrepancy between ∂ξ and x. For such (ξ, x) let Gξ,x = (V, Aξ,x ) be a directed graph with vertex set V and arc set Aξ,x = A∗ξ ∪ Bξ∗ ∪ Cx consisting of three disjoint parts: A∗ξ = {a | a ∈ A, ξ(a) < c(a)}, Bξ∗ = {a | a ∈ A, c(a) < ξ(a)} (a: reorientation of a), Cx = {(u, v) | u, v ∈ V, u = v, x − χu + χv ∈ B},

(10.51)

and define S − = {v | x(v) < ∂ξ(v)}. In order to reduce the discrepancy ||x − ∂ξ||1 = v∈V |x(v) − ∂ξ(v)|, the algorithm augments a unit flow along a shortest path P from S + to S − (shortest with respect to the number of arcs) and modifies ξ to ξ given by ⎧ ⎨ ξ(a) + 1 (a ∈ P ∩ A∗ξ ), ξ(a) − 1 (a ∈ P ∩ Bξ∗ ), ξ(a) = ⎩ ξ(a) (otherwise). S + = {v | x(v) > ∂ξ(v)},

Obviously, ξ satisfies the capacity constraint (10.49). The algorithm also updates the base x to  x=x− {χu − χv | (u, v) ∈ P ∩ Cx }, (10.52) which remains a base belonging to B; see Note 10.33. For the initial vertex s ∈ S + of the path P , either ∂ξ(s) increases or x(s) decreases by one and hence |x(s) − ∂ξ(s)| reduces by one. A similar result is true for the terminal vertex of P in S − , whereas x(v) = ∂ξ(v) is kept invariant at every inner vertex v of P . Therefore, each augmentation along a shortest path decreases ||x − ∂ξ||1 by two. Repeating this process until the source S + and consequently the sink S − become empty, the algorithm constructs a pair (ξ, x) with ∂ξ = x. Then ξ is a feasible flow satisfying both (10.49) and (10.50).

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Successive shortest path algorithm for finding a feasible integer flow S0: Find ξ ∈ ZA satisfying (10.49) and x ∈ B. S1: If S + is empty, then stop (ξ is a feasible flow). S2: If there is no path from S + to S − , then stop (no feasible flow exists). S3: Let P⎧be a shortest path from S + to S − and, for each arc a ∈ P , set ⎨ a ∈ A∗ξ ⇒ ξ(a) := ξ(a) + 1, (a: reorientation of a), a ∈ Bξ∗ ⇒ ξ(a) := ξ(a) − 1 ⎩ a ∈ Cx ⇒ x(∂ + a) := x(∂ + a) − 1, x(∂ − a) := x(∂ − a) + 1 and go to S1. Note 10.33. The updated vector x in (10.52) remains a base, i.e., x ∈ B, by Proposition 9.23 with f = δB (indicator function of B). Let G(x, x) be the bipartite graph, as defined in section 9.5.2. All the arcs have zero weight. Hence, (x, x) meets the unique-min condition if and only if G(x, x) has a unique perfect matching. The latter condition holds in the algorithm because P is chosen to be a shortest path, whereas Proposition 9.23 says that the unique-min condition implies x ∈ B. Note 10.34. Instead of augmenting a unit flow along P it is more efficient to augment as much as possible. The maximum admissible amount is given by δ = min{c(a) | a ∈ P } with ⎧ ⎨ c(a) − ξ(a) (a ∈ A∗ξ ), ξ(a) − c(a) (a ∈ Bξ∗ , a ∈ A), c(a) = ⎩ cB (x, v, u) (a = (u, v) ∈ Cx ), where cB (·, ·, ·) means the exchange capacity defined in (10.4). It can be shown that  x=x−δ {χu − χv | (u, v) ∈ P ∩ Cx } stays in B; see Lemma 4.5 of Fujishige [65]. The successive shortest path algorithm can be adapted to finding a real-valued feasible flow to a nonintegral submodular flow problem. For a polynomial complexity bound of the algorithm, it is important to choose, in the case of multiple candidates, an appropriate shortest path with reference to some lexicographic ordering. The successive shortest path algorithm can be generalized for optimal flow problems involving cost functions; see [65], Fujishige–Iwata [66], and Iwata [98] for such algorithms for MSFP1 (without Mconvex cost), whereas such an algorithm for the integer-flow version of MSFP2 (with M-convex cost) is given in Moriguchi–Murota [132]. Note 10.35. A number of algorithms are available for finding a feasible submodular flow; e.g., Fujishige [59], Frank [56], Tardos–Tovey–Trick [199], and Fujishige–Zhang [70]. The reader is referred to Fujishige [65], Fujishige–Iwata [66], and Iwata [98] for expositions.

10.4.3

Cycle-Canceling Algorithm

For the M-convex submodular integer-flow problem with linear arc cost, we have seen in section 9.5 that a feasible flow is optimal if and only if there exists no negative

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cycle in an auxiliary network (Theorem 9.20) and that a nonoptimal flow can be improved by augmenting a flow along a suitably chosen negative cycle (Theorem 9.22). These two facts suggest the following cycle-canceling algorithm, which works on the auxiliary network (Gξ , ξ ) introduced in section 9.5. Cycle-canceling algorithm for MSFP2 (integer flow) S0: Find a feasible integer flow ξ. S1: If (Gξ , ξ ) has no negative cycle, then stop (ξ is an optimal flow). S2: Let Q be a negative cycle with the smallest number of arcs. S3: Modify ξ to ξ of (9.75) and go to S1. The objective function Γ2 decreases monotonically by Theorem 9.22. This property alone does not ensure finite termination in general, but it does if Γ2 is integer valued and bounded from below. In the special case with dom f ⊆ {0, 1}V , which corresponds to the valuated matroid intersection problem (Example 8.28), some variants of the cycle-canceling algorithm are known to be strongly polynomial (see Murota [136]). Cycle-canceling algorithms can also be designed for problems with real-valued flow on the basis of Theorem 9.18. Note 10.36. For MSFP1 (submodular flow problem with linear arc cost and without M-convex cost), a number of cycle-canceling algorithms are proposed, including Fujishige [59], Cui–Fujishige [27], Zimmermann [222], Wallacher–Zimmermann [210], and Iwata–McCormick–Shigeno [103].

10.4.4

Primal-Dual Algorithm

A primal-dual algorithm for the M-convex submodular flow problem is described. The algorithm maintains a pair of flow and potential and modifies them to optimality. We deal with the case of linear arc cost with dual integrality. M-convex submodular flow problem MSFP2  Minimize Γ2 (ξ) = γ(a)ξ(a) + f (∂ξ) (10.53) a∈A

subject to c(a) ≤ ξ(a) ≤ c(a)

(a ∈ A),

∂ξ ∈ dom f, ξ(a) ∈ R

(10.54) (10.55)

(a ∈ A).

(10.56)

Here, c : A → R ∪ {+∞}, c : A → R ∪ {−∞}, f ∈ M[R → R|Z], and γ : A → Z (see (9.67)). The feasibility of the problem is also assumed. By Theorems 9.14 and 9.15 as well as (9.31), (9.32), and (9.65), a feasible flow ξ : A → R is optimal if and only if there exists an integer-valued potential p : V → Z such that γp (a) > 0 =⇒ ξ(a) = c(a),

(10.57)

γp (a) < 0 =⇒ ξ(a) = c(a), ∂ξ ∈ B(gp ),

(10.58) (10.59)

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where γp : A → Z is the (integer-valued) reduced cost defined by γp (a) = γ(a) + p(∂ + a) − p(∂ − a)

(a ∈ A),

(10.60)

gp : 2V → R ∪ {+∞} is the submodular set function derived from g = f • ∈ L[Z|R → R] by gp (X) = g  (p; χX ) = g(p + χX ) − g(p)

(X ⊆ V )

with gp (V ) = 0 by (9.51), and B(gp ) is the base polyhedron (M-convex polyhedron) associated with gp . The algorithm maintains a pair (ξ, p) ∈ RA × ZV of a feasible flow ξ and an integer-valued potential p that satisfies (10.59) and repeats modifying (ξ, p) to increase the set of arcs satisfying (10.57) and (10.58). We say an arc is in kilter with respect to (ξ, p) if it satisfies (10.57) and (10.58) and out of kilter otherwise. Note that exactly one of (10.57) and (10.58) fails for an out-of-kilter arc. An initial (ξ, p) satisfying (10.59) can be found as follows. For any feasible flow ξ we consider a graph Gξ = (V, Cξ ) with vertex set V and arc set Cξ = {(u, v) | u, v ∈ V, u = v, ∃ α > 0 : ∂ξ − α(χu − χv ) ∈ dom f } and define the length of arc (u, v) as f  (∂ξ; −χu + χv ), which is an integer by the assumed dual integrality (Note 9.19). By solving a shortest path problem we can find an integral vector p such that p(v) − p(u) ≤ f  (∂ξ; −χu + χv )

(u, v ∈ V ),

(10.61)

which implies ∂ξ ∈ arg min f [−p] = B(gp ) by (9.64) and (9.65). To classify out-of-kilter arcs we define Dξ+ (v) = {a ∈ δ + v | γp (a) < 0, ξ(a) < c(a)}

(v ∈ V ),

Dξ− (v)

(v ∈ V ),



= {a ∈ δ v | γp (a) > 0, ξ(a) > c(a)}

Dξ (v) =

Dξ+ (v)



Dξ− (v)

(v ∈ V ),

where the dependence on p is implicit in the notation. Note that Dξ+ (v) is the set of arcs leaving v for which (10.58) fails, Dξ− (v) is the set of arcs entering v for which (10.57) fails, and {Dξ (v) | v ∈ V } gives a partition of the set of out-of-kilter arcs. If Dξ (v) = ∅ for all v ∈ V , condition (i) of (POT) is satisfied and the current (ξ, p) is optimal. Otherwise, the algorithm picks62 any v  ∈ V with nonempty Dξ (v  ) and tries to meet the conditions (10.57) for a ∈ Dξ− (v  ) and (10.58) for a ∈ Dξ+ (v  ) by changing the flow to ξ  . The flow ξ  is determined by solving the following maximum submodular flow problem on G = (V, A) with a more restrictive capacity constraint c (a) ≤ ξ  (a) ≤ c (a) with 62 The original primal-dual algorithm picks an out-of-kilter arc and tries to meet (10.57) and (10.58) for that arc. The present strategy to pick a vertex is to improve the worst-case complexity bound.

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315

⎧ ⎨ ξ(a) (γp (a) > 0), c (a) = c(a) (γp (a) = 0), ⎩ c(a) (γp (a) < 0)

(γp (a) > 0), (γp (a) = 0), (γp (a) < 0),

(10.62)

for each a ∈ A. Maximum submodular flow problem maxSFP Maximize

ξ  (Dξ+ (v  )) − ξ  (Dξ− (v  )) 

subject to c (a) ≤ ξ (a) ≤ c (a) 



(10.63) (a ∈ A),



∂ξ ∈ B(gp ), (a ∈ A), ξ  (a) ∈ R

(10.64) (10.65) (10.66)

where ξ  : A → R is the variable to be optimized. Since the capacity interval [c (a), c (a)] is included in the original capacity interval [c(a), c(a)], any feasible flow in maxSFP is feasible in MSFP2 . Note also that maxSFP has a feasible flow ξ  = ξ. In maximizing the objective function   ξ  (a) − ξ  (a) ξ  (Dξ+ (v  )) − ξ  (Dξ− (v  )) = a∈Dξ+ (v  )

a∈Dξ− (v  )

it is intended to meet the conditions (10.58) and (10.57) by increasing the flow in a ∈ Dξ+ (v  ) to the upper capacity and decreasing the flow in a ∈ Dξ− (v  ) to the lower capacity. The maximum submodular flow problem is a special case of the feasibility problem for the submodular flow problem and a number of efficient algorithms are available for it (see section 10.4.2). Let ξ  be an optimal solution to maxSFP above. If Dξ (v  ) is empty, the algorithm updates ξ to ξ  without changing p. The condition (10.59) is maintained because of (10.65). If Dξ (v  ) is nonempty, the algorithm finds a minimum cut W ⊆ V (explained later) and updates p to p  = p + χW

(10.67)

as well as ξ to ξ  . The condition (10.59) is maintained, as is shown in Proposition 10.38 below. The primal-dual algorithm for the M-convex submodular flow problem with a linear arc cost (MSFP2 ) with dual integrality is summarized as follows. Primal-dual algorithm for MSFP2 with dual integrality S0: Find (ξ, p) ∈ RA × ZV satisfying (10.54) and (10.59). S1: If Dξ (v) = ∅ for all v ∈ V , then stop ((ξ, p) is optimal). S2: Take any v  ∈ V with Dξ (v  ) = ∅ and solve maxSFP to obtain ξ  . S3: If Dξ (v  ) = ∅, then find a minimum cut W and set p := p + χW . S4: Set ξ := ξ  and go to S1. It remains to explain the minimum cut for maxSFP. For W ⊆ V containing v  , we define the cut capacity ν(W ) by

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316

Chapter 10. Algorithms ˆ G:

G: Dξ− (v  )

Dξ+ (v  ) v

z : γp < 0 ξ 0 ξ>c

in kilter

z

vˆ

Dξ− (v  )

:

a0 6  v

γp < 0 ξ 0 ξ>c

in kilter

ˆ at v  . Figure 10.1. Structure of G and G

ν(W ) = c (Δ− W \ Dξ− (v  )) + c (Dξ+ (v  ) \ Δ+ W ) −c (Δ+ W \ Dξ+ (v  )) − c (Dξ− (v  ) \ Δ− W ) +g(p + χW ) − g(p), where Δ+ W and Δ− W mean the sets of arcs leaving W and entering W , respectively, as in (9.14) and (9.15). The following proposition states that the flow value ξ  (Dξ+ (v  ))− ξ  (Dξ− (v  )) is bounded by ν(W ) for any W ⊆ V with v  ∈ W and that this bound is tight for some W , which is referred to as a minimum cut in the above. A minimum cut can be found with the aid of an appropriately defined auxiliary network. Proposition 10.37. In the maximum submodular flow problem maxSFP, we have max{ξ  (Dξ+ (v  )) − ξ  (Dξ− (v  )) | (10.64), (10.65), (10.66)} = min{ν(W ) | v  ∈ W ⊆ V }.

(10.68)

For a maximum flow ξ  and a minimum cut W , we have ∂ξ  (W ) = g(p + χW ) − g(p), (a ∈ (Δ+ W \ Dξ+ (v  )) ∪ (Dξ− (v  ) \ Δ− W )), ξ  (a) = c (a)

(10.69) (10.70)

(a ∈ (Δ− W \ Dξ− (v  )) ∪ (Dξ+ (v  ) \ Δ+ W )).

(10.71)

ξ  (a) = c (a)

Proof. We prove this by applying Theorem 9.13 (max-flow min-cut theorem) to a ˆ = (Vˆ , A), ˆ which is obtained from maximum submodular flow problem on a graph G  G = (V, A) by a local modification at v illustrated in Fig. 10.1. The vertex v  in G is split into two vertices, v  and vˆ , and a new arc a0 = (v  , vˆ ) is introduced; Vˆ = V ∪ {ˆ v  } and Aˆ = A ∪ {a0 }. The initial vertex of a ∈ Dξ+ (v  ) is changed to vˆ and the terminal vertex of a ∈ Dξ− (v  ) is changed to vˆ . For W ⊆ Vˆ we denote by ˆ + W and Δ ˆ − W the sets of arcs leaving and entering W , respectively, in G. ˆ Δ ˆ= The problem maxSFP is equivalent to maximizing the flow ξ  (a0 ) in a0 in G ˆ where the conservation of flow at vˆ (i.e., ∂ξ  (ˆ v  ) = 0) is assumed and no (Vˆ , A),

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capacity constraint is imposed on a0 . Note that ξ  (a0 ) = ξ  (Dξ+ (v  ))−ξ  (Dξ− (v  )) as a consequence of the flow conservation at vˆ . As for cuts, we note the correspondence ˆ + W and W ⊆ V with v  ∈ W and observe the between W ⊆ Vˆ with a0 ∈ Δ identities ˆ + W = (Δ+ W \ D+ (v  )) ∪ (D− (v  ) \ Δ− W ) ∪ {a0 }, Δ ξ

ξ

ˆ − W = (Δ− W \ D− (v  )) ∪ (D+ (v  ) \ Δ+ W ) Δ ξ ξ for such a W . Then we obtain (10.68) from (9.61) in Theorem 9.13. Note that g(p + χW ) − g(p) = gp (W ) in ν(W ) corresponds to ρ in (9.61). Finally, (10.69), (10.70), and (10.71) are shown in (9.62). The condition (10.59) is maintained when (ξ, p) is modified to (ξ  , p ). Proposition 10.38. ∂ξ  ∈ B(gp ) for an optimum flow ξ  in maxSFP and potential p = p + χW with a minimum cut W . Proof. It follows from (10.65), (10.69), and discrete midpoint convexity (7.7) that ∂ξ  (X) = ∂ξ  (X ∪ W ) + ∂ξ  (X ∩ W ) − ∂ξ  (W ) ≤ g(p + χX∪W ) + g(p + χX∩W ) − g(p) − g(p + χW ) (X ⊆ V ). ≤ g(p + χW + χX ) − g(p + χW ) = gp (X) This shows ∂ξ  ∈ B(gp ). The following proposition shows the key properties for the correctness and complexity of the primal-dual algorithm. Proposition 10.39. (1) The set of out-of-kilter arcs is nonincreasing. (2) For each arc a, |γp (a)| is nonincreasing while the arc stays out of kilter. (3) The potential is changed in at most |V | iterations and, each time the potential is changed, the value of maxa∈Dξ (v ) |γp (a)| decreases at least by one. (4) Each time (ξ, p) is changed, the value of ) A  ) N= (10.72) max |γp (a)|)) v ∈ V, Dξ (v) = ∅ a∈Dξ (v) v

decreases at least by one. Therefore, the primal-dual algorithm terminates in at most N0 iterations, where N0 denotes the value of N at step S0. Proof. The reader is referred to the kilter diagram in Fig. 9.1. (1) We show that an in-kilter arc a with respect to (ξ, p) remains in kilter in updating (ξ, p). It follows from ξ  (a) ∈ [c (a), c (a)] and (10.62) that a remains in kilter with respect to (ξ  , p). Suppose that p is updated to p = p + χW . Since ⎧ ⎨ +1 (a ∈ Δ+ W ), −1 (a ∈ Δ− W ), γp (a) − γp (a) = (10.73) ⎩ 0 (otherwise),

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Chapter 10. Algorithms

we may assume that a is in kilter with respect to (ξ  , p) and (i) a ∈ Δ+ W \ Dξ+ (v  ) or (ii) a ∈ Δ− W \ Dξ− (v  ). In case (i) we have ξ  (a) = c (a) from (10.70), whereas γp (a) > 0 ⇒ γp (a) ≥ 0 ⇒ c (a) = c(a), γp (a) < 0 ⇒ γp (a) < 0 ⇒ c (a) = ξ(a) = c(a). In case (ii) we have ξ  (a) = c (a) from (10.71), whereas γp (a) > 0 ⇒ γp (a) > 0 ⇒ c (a) = ξ(a) = c(a), γp (a) < 0 ⇒ γp (a) ≤ 0 ⇒ c (a) = c(a). Thus the conditions (10.57) and (10.58) are preserved in either case. (2) By (10.73), it suffices to show that, if (i) γp (a) > 0, a ∈ Δ+ W , or (ii) γp (a) < 0, a ∈ Δ− W , then a is in kilter with respect to (ξ  , p ). In case (i), we have a ∈ Δ+ W \ Dξ+ (v  ), from which follows ξ  (a) = c (a) = c(a) by (10.70). Since γp (a) > 0, a is in kilter with respect to (ξ  , p ), and similarly for case (ii). (3) If the potential does not change, we have Dξ (v  ) = ∅ as well as Dξ (v) ⊆ Dξ (v) for all v ∈ V by (1). Therefore, the potential must be updated in |V | iterations. Suppose now that p is changed to p . An arc a ∈ Dξ+ (v  ) \ Δ+ W is in kilter with respect to (ξ  , p ), since γp (a) ≤ γp (a) < 0 and ξ  (a) = c (a) = c(a) by (10.71). Similarly, a ∈ Dξ− (v  ) \ Δ− W is in kilter with respect to (ξ  , p ) because of (10.70). For a ∈ Dξ+ (v  )∩Δ+ W , we have |γp (a)| = |γp (a)|−1 from γp (a) = γp (a)+ 1 and γp (a) < 0. Similarly, for a ∈ Dξ− (v  ) ∩ Δ− W , we have |γp (a)| = |γp (a)| − 1 from γp (a) = γp (a) − 1 and γp (a) > 0. Therefore, maxa∈Dξ (v ) |γp (a)| decreases at least by one. (4) When the potential changes, N decreases because of (3). When the potential remains invariant, N decreases because of Dξ (v  ) = ∅. Primal-dual algorithms can also be designed for problems without dual integrality by using real-valued potential functions. Note 10.40. The framework of the primal-dual algorithm for MSFP1 (submodular flow problem without M-convex cost) was established in Frank [55] and Cunningham– Frank [30]. See Fujishige [65], Fujishige–Iwata [66], and Iwata [98] for expositions.

10.4.5

Conjugate Scaling Algorithm

With the use of conjugate scaling of the M-convex cost function, the primal-dual algorithm is enhanced to a polynomial-time algorithm. We continue to deal with the M-convex submodular flow problem MSFP2 with dual integrality; i.e., we assume γ : A → Z and f ∈ M[R → R|Z]. First we explain the intuition behind the cost-scaling algorithm for the submodular flow problem MSFP1 without M-convex function (see section 9.2 for MSFP1 ). As Proposition 10.39 (4) shows, the time complexity of the primal-dual algorithm depends essentially on |γp (a)| = |γ(a) + p(∂ + a) − p(∂ − a)|.

(10.74)

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319

Motivated by this fact, we consider MSFP1 with a new objective function   γ(a)  ξ(a), (10.75) α a∈A

where α is a positive integer representing cost scaling and · means rounding up to the nearest integer. It is expected (or hoped) that such a scaling will result in smaller values of (10.74) and hence in an improvement in the computation time of the algorithm. On the other hand, the scaled problem with (10.75) is fairly close to the original problem, since α γ(a)/α  γ(a), and, therefore, the solution to the scaled problem is likely to be a good approximation that can be used as an initial solution in solving the original problem by the primal-dual algorithm. The scaling algorithm embodies the above idea by starting with a large α and successively halving α until α = 1. When an M-convex function f is involved, as in MSFP2 , it is natural to try a scaling of the form f (·)/α . This approach, however, does not seem to work in general, since f (·)/α is not necessarily M-convex for an M-convex function f . Conjugate scaling is a kind of scaling operation compatible with M-convexity. Let f : RV → R∪{+∞} be a polyhedral convex function with dual integrality in the sense that the conjugate function g = f • has integrality (6.75). Then we have f (x) = sup{ p, x − g(p) | p ∈ ZV }

(x ∈ RV ),

(10.76)

where the supremum is taken over integer points. Replacing g(p) with g α (p) = g(αp)/α (p ∈ ZV ) in this expression we define ) A  ) 1 α V ) f (x) = sup p, x − g(αp)) p ∈ Z (x ∈ RV ), (10.77) α which we call the conjugate scaling of f with scaling factor α ∈ Z++ . Note that f α is again a dual-integral polyhedral convex function. It is easy to see that f α (x) ≤

1 f (x) α

(x ∈ RV )

and that domR f α = domR f provided f α > −∞. Figure 10.2 illustrates the conjugate scaling of a univariate function f with α = 2. Proposition 10.41. For a dual-integral polyhedral M-convex function f ∈ M[R → R|Z], we have f α ∈ M[R → R|Z] provided f α > −∞. Proof. We have f • = g ∈ L[Z|R → R] and hence g α ∈ L[Z → R] by Theorem 7.10 (2). Therefore, f α = (g α )• ∈ M[R → R|Z] by (8.10). We are now in the position to present the conjugate scaling algorithm. Initially the algorithm finds a feasible flow ξ0 and an integer-valued potential p0 satisfying (10.59) and applies the conjugate scaling to the objective function rewritten as    γ(a)ξ(a)+f (∂ξ) = γp0 (a)ξ(a)+f [−p0 ](∂ξ) = γ˜ (a)ξ(a)+ f˜(∂ξ), Γ2 (ξ) = a∈A

a∈A

a∈A

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Chapter 10. Algorithms

f (x)

8 6

8 6

αf α (x) 6

−3 −2 −1

g(p) αg α ( αp )

6

4

4

2

2

0

1

2

3 x

−1

0

1

2

3

4 p

Figure 10.2. Conjugate scaling f α and scaling g α for α = 2.

where γ˜ = γp0 and f˜ = f [−p0 ]. We denote the conjugate scaling of f˜ by f˜α and put g˜ = f˜• . Note that f˜ ∈ M[R → R|Z] and g˜(p) = g(p + p0 ),

g˜α (p) =

1 g(αp + p0 ) α

(p ∈ ZV ),

where we regard g˜ as a member of L[Z → R]. Recall the notation n = |V |. Conjugate scaling algorithm for MSFP2 with dual integrality S0: Find a feasible flow ξ0 ∈ RA and an integer-valued potential p0 ∈ ZV satisfying (10.59) and define γ˜, f˜, and g˜ accordingly. Set p∗ := 0, K := maxa∈A |˜ γ (a)|, α := 2log2 K . S1: If α < 1, then stop ((ξ, p0 + p∗ ) is optimal). S2: Find an integer vector p ∈ ZV that minimizes g˜α (p) − p, ∂ξ subject to 2p∗ ≤ p ≤ 2p∗ + (n − 1)1. S3: Solve MSFP2 for (γ, f ) = ( ˜ γ /α , f˜α ) by the primal-dual algorithm starting with (ξ, p) to obtain an optimal (ξ ∗ , p∗ ) ∈ RA × ZV . S4: Set ξ := ξ ∗ and α := α/2 and go to S1. The correctness of the algorithm is ensured by Theorem 7.18 (L-proximity theorem), which implies that the minimizer p found in step S2 under the restriction 2p∗ ≤ p ≤ 2p∗ + (n − 1)1 is in fact a global minimizer of g˜α (p) − p, ∂ξ . Hence, the condition ∂ξ ∈ B(˜ g α p ) = arg min f˜α [−p] for (10.59) is maintained. In step S2, the minimizer p can be found by the L-convex function minimization algorithms of section 10.3, where the number of evaluations of g˜α is bounded by a polynomial in n. Given f , an evaluation of g˜α amounts to minimizing a polyhedral M-convex function, since g(p) = sup{ p, x − f (x) | x ∈ RV }. If f has primal

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321

integrality, i.e., if f ∈ M[Z|R → R], then g(p) = sup{ p, x −f (x) | x ∈ ZV }, which can be computed by the algorithms in section 10.1. In step S3, the number of iterations (updates of (ξ, p)) within the primal-dual algorithm is bounded by n2 . Denote by pα the value of p at the beginning of step S3 and put p2α = p∗ , γ α = ˜ γ /α , and γ 2α = ˜ γ /(2α) . Then we have 2γ 2α − 1 ≤ γ α ≤ 2γ 2α ,

2p2α ≤ pα ≤ 2p2α + (n − 1)1,

from which follows ) ) α )[γ (a) + pα (∂ + a) − pα (∂ − a)] − 2[γ 2α (a) + p2α (∂ + a) − p2α (∂ − a)]) ≤ n. This means that at the beginning of step S3 we have ) α ) )γ (a) + pα (∂ + a) − pα (∂ − a)) ≤ n for every out-of-kilter arc a and therefore N in (10.72) is bounded by n2 . Obviously, steps S1 through S4 are repeated log2 K times. For the value of K we have (10.78) K ≤ max |γ(a)| + max max |f  (x; −χu + χv )| x

a∈A

u,v∈V

if the initial potential p0 in step S0 is computed from a shortest path on the graph Gξ = (V, Cξ ), as explained in section 10.4.4, where in the second term on the right-hand side we consider only those x, u, v for which f  (x; −χu + χv ) is finite. If f ∈ M[Z|R → R], the second term can be bounded as max max |f  (x; −χu + χv )| ≤ 2 max |f (x)|, x

u,v∈V

x∈dom f

which implies K ≤ max |γ(a)| + 2 max |f (x)|. a∈A

x∈dom f

(10.79)

Bibliographical Notes The tie-breaking rule (10.2) for M-convex function minimization, as well as Proposition 10.2, is due to Murota [148]. Variants of steepest descent algorithms are reported in Moriguchi–Murota–Shioura [133] with some computational results. Scaling algorithms for M-convex function minimization, including the one described in section 10.1.2, were considered first in [133], although the proposed algorithms run in polynomial time only for a subclass of M-convex functions that are closed under scaling. A polynomial-time scaling algorithm for general M-convex functions is given by Tamura [197]. The domain reduction algorithm in section 10.1.3 is due to Shioura [190] and its extension to quasi M-convex functions is observed in Murota– Shioura [154]. The domain reduction scaling algorithm in section 10.1.4, with its extension to quasi M-convex functions, is due to Shioura [192]. Minimizing an Mconvex function on {0, 1}-vectors is equivalent to maximizing a matroid valuation, for which a greedy algorithm of Dress–Wenzel [41] works; see also section 5.2.4 of Murota [146].

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Chapter 10. Algorithms

The literature of submodular function minimization was described in Note 10.10. The algorithmic framework expounded in section 10.2.1 is due to Cunningham [28], [29] as well as Bixby–Cunningham–Topkis [15]. The algorithm in section 10.2.2 is by Schrijver [182], whereas an earlier version of this algorithm based on partial orders associated with extreme bases (presented at the Workshop on Polyhedral and Semidefinite Programming Methods in Combinatorial Optimization, Fields Institute, November 1–6, 1999) is described in Murota [147]. The algorithm in section 10.2.3 is due to Iwata–Fleischer–Fujishige [102]. Improvements on those algorithms in terms of time complexity were made by Fleischer–Iwata [50] and Iwata [100]. See McCormick [127] for a detailed survey on submodular function minimization. Note 10.12 was communicated by K. Nagano, Note 10.14 is based on [182], and Proposition 10.28 was communicated by S. Iwata. Favati–Tardella [49] proposes a weakly polynomial algorithm for submodular integrally convex function minimization. This is the first polynomial algorithm for L-convex function minimization, when translated through the equivalence between submodular integrally convex functions and L -convex functions. The steepest descent algorithm for L-convex function minimization in section 10.3.1 is given in Murota [145]. The tie-breaking rule (10.33), as well as Proposition 10.31, is due to Murota [148]. The steepest descent scaling algorithm in section 10.3.2 is due to S. Iwata (presented at Workshop on Matroids, Matching, and Extensions, University of Waterloo, December 6–11, 1999), where step S1 is performed not by the steepest descent algorithm but by the algorithm in section 10.3.3. The framework of M-convex submodular flow problems is advanced by Murota [142]. The successive shortest path algorithm for a feasible flow described in section 10.4.2 originates in Fujishige [59] and the present form is due to Frank [56]. The cycle-canceling algorithm of section 10.4.3 is devised in [142] as a proof of the negative-cycle criterion for optimality (Theorem 9.20). In the special case of valuated matroid intersection, the algorithm can be polished to a strongly polynomial algorithm (Murota [136]); see also Note 10.36. The primal-dual algorithm of section 10.4.4 is due to Iwata–Shigeno [105]; see also Note 10.40. A strongly polynomial primal-dual algorithm for the valuated matroid intersection problem is given in [136]. The conjugate scaling algorithm of section 10.4.5 is due to [105]. A scaling algorithm for a subclass of the M-convex submodular flow problem is given by Moriguchi–Murota [132]. Capacity scaling algorithms for submodular flow problems (without M-convex costs) are given in Iwata [97] and Fleischer–Iwata–McCormick [51]. For other algorithms for submodular flow problems (without M-convex costs), see the book of Fujishige [65] and surveys of Fujishige–Iwata [66] and Iwata [98].

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Chapter 11

Application to Mathematical Economics

This chapter presents an application of discrete convex analysis to a subject in mathematical economics: competitive equilibria in economies with indivisible (or discrete) commodities. For economies consisting of continuous commodities, represented by real-valued vectors, a rigorous mathematical framework was established around 1960 on the basis of convexity, compactness, and fixed-point theorems. For indivisible commodities, however, no general mathematical framework seems to have been established. Such a framework, if any, should embrace both convexity and discreteness; the present theory of discrete convex analysis appears to be a promising candidate for it. It is shown that, in an Arrow–Debreu type model of competitive economies with indivisible commodities, an equilibrium exists under the assumption of the M -concavity of consumers’ utility functions and the M -convexity of producers’ cost functions. Moreover, the equilibrium prices form an L -convex polyhedron, and, therefore, they have maximum and minimum elements. The conjugacy between M-convexity and L-convexity corresponds to the relationship between commodities and prices.

11.1

Economic Model with Indivisible Commodities

As an application of discrete convex analysis we deal with competitive equilibria in economies with a number of indivisible commodities and money. Indivisible commodities mean commodities (goods) whose quantities are represented by integers, such as houses, cars, and aircraft, whereas money is a real number representing the aggregation of the markets of other commodities. We consider an economy (of Arrow–Debreu type) with a finite set L of producers, a finite set H of consumers, a finite set K of indivisible commodities, and a perfectly divisible commodity called money. Productions of producers and consumptions of consumers are integer-valued vectors in ZK representing the numbers of indivisible commodities that they produce or consume. Here producers’ outputs are represented by positive numbers, while negative numbers are interpreted as inputs to them, and consumers’ inputs are represented by positive numbers, 323

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Chapter 11. Application to Mathematical Economics

while negative numbers are interpreted as outputs from them. Given a price vector p = (p(k) : k ∈ K) ∈ RK of commodities, each producer (assumed to be male) independently schedules a production in order to maximize his profit, each consumer (assumed to be female) independently schedules a consumption to maximize her utility under the budget constraint, and all agents exchange commodities by buying or selling them through money. An important feature of this model is that the independent agents take the price as granted; i.e., they assume that their individual behaviors do not affect the price. Such an economy is called a competitive economy. We assume that a producer l ∈ L is described by his cost function Cl : ZK → R ∪ {+∞}, whose value is expressed in units of money. He wishes to maximize the profit p, y − Cl (y) in determining his production y = yl ∈ ZK . This means that yl is chosen from the supply set Sl (p) = arg max ( p, y − Cl (y)) y∈ZK

(p ∈ RK ),

(11.1)

K

where the function Sl : RK → 2Z is called the supply correspondence. Accordingly, the profit function πl : RK → R is defined by πl (p) = max ( p, y − Cl (y)) y∈ZK

(p ∈ RK ).

(11.2)

To avoid possible technical complications irrelevant to discreteness issues, we assume that dom Cl is a bounded subset of ZK for each l ∈ L. This guarantees, for instance, that Sl (p) is nonempty for any p. Each consumer h ∈ H has an initial endowment of indivisible commodities and money, represented by a vector (x◦h , m◦h ) ∈ ZK × R+ , where x◦h (k) denotes the number of the commodity k ∈ K and m◦h the amount of money in her initial endowment. Consumers share in the profits of the producers. We denote by θlh the share of the profit of producer l owned by consumer h, where  θlh = 1 (l ∈ L), θlh ≥ 0 (l ∈ L, h ∈ H). (11.3) h∈H

Thus, consumer h gains an income βh (p) = p, x◦h + m◦h +



θlh πl (p)

(p ∈ RK ),

(11.4)

l∈L

where βh : RK → R, and accordingly her schedule (x, m) = (xh , mh ) should belong to her budget set Bh (p) = {(x, m) ∈ ZK × R+ | p, x + m ≤ βh (p)}.

(11.5)

¯h : ZK × R → We assume that a consumer h is associated with a utility function U R ∪ {−∞} that is quasi linear in money; namely, ¯h (x, m) = Uh (x) + m U

((x, m) ∈ ZK × R)

(11.6)

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11.1. Economic Model with Indivisible Commodities

money m

325

6 indifference curve ¯h ) (contour of utility U

slope = −p

Bh (p) Dh (p)

indivisible commodities x

Figure 11.1. Consumer’s behavior .

¯h under the budwith a function63 Uh : ZK → R ∪{−∞}. Consumer h maximizes U get constraint; that is, (x, m) = (xh , mh ) is a solution to the following optimization problem: Maximize

Uh (x) + m

(x, m) ∈ Bh (p)

subject to

(see Fig. 11.1). Under the assumption that dom Uh is bounded64 and m◦h is sufficiently large, we can take mh = βh (p) − p, xh

(≥ 0)

(11.7)

to reduce the above problem to an unconstrained optimization problem: Maximize Uh (x) − p, x . This means that xh is chosen from the demand set Dh (p) = arg max (Uh (x) − p, x ) x∈ZK

(p ∈ RK ).

(11.8)

K

The function Dh : RK → 2Z is called the demand correspondence. A tuple ((xh | h ∈ H), (yl | l ∈ L), p), where xh ∈ ZK , yl ∈ ZK , and p ∈ RK , is called an equilibrium or a competitive equilibrium if xh ∈ Dh (p) 63 In

(h ∈ H),

(11.9)

economic terminology, Uh is called the reservation value function, although we refer to it as the utility function in this book. 64 The boundedness of dom U is a natural assumption because no one can consume an infih nite number of indivisible commodities. This assumption is also convenient for concentrating on discreteness issues in our discussion.

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326

Chapter 11. Application to Mathematical Economics (l ∈ L), yl ∈ Sl (p)    xh = x◦h + yl , h∈H

h∈H

(11.10) (11.11)

l∈L

p ≥ 0.

(11.12)

That is, each agent achieves what he or she wishes to achieve, the balance of supply and demand holds, and an equilibrium price vector is nonnegative. Denoting the total initial endowment of indivisible commodities by  x◦h , (11.13) x◦ = h∈H

we can rewrite the supply-demand balance (11.11) as   x◦ = xh − yl . h∈H

(11.14)

l∈L

On eliminating xh and yl using (11.9) and (11.10), we see that p ∈ RK + is an equilibrium price if and only if   Dh (p) − Sl (p), (11.15) x◦ ∈ h∈H

l∈L

where the right-hand side is a Minkowski sum in ZK . It is noted that money balance    mh = m◦h − Cl (yl ) (11.16) h∈H

h∈H

l∈L

is implied by (11.11) with (11.3), (11.4), (11.7), and πl (p) = p, yl − Cl (yl ). We are concerned with mathematical properties of equilibria, rather than their economic-theoretical significance. A most fundamental question would be as follows: When does an equilibrium exist? Namely, the first problem we should address is this: Problem 1: Give a (sufficient) condition for the existence of an equilibrium in terms of utility functions Uh and cost functions Cl . The conditions (11.9) and (11.10) for an equilibrium are given in terms of demand correspondences Dh and supply correspondences Sl without explicit reference to utility functions Uh and cost functions Cl . This motivates the following: Problem 2: Give a (sufficient) condition for the existence of an equilibrium in terms of demand correspondences Dh and supply correspondences Sl . When an equilibrium exists, we may be interested in its structure: Problem 3: Investigate the structure of the set of equilibria. A more specific problem in this category is as follows: Do the maximum and minimum exist among equilibrium price vectors? We shall answer the above problems with the use of concepts and results in discrete convex analysis. Our answers are the following.

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327

(1) An equilibrium exists if Uh (h ∈ H) are M -concave functions and Cl (l ∈ L) are M -convex functions (Theorems 11.13 and 11.14). (2) An equilibrium exists if Dh (p) (h ∈ H) and Sl (p) (l ∈ L) are M -convex sets for each p (Theorem 11.15). (3) The set P ∗ of the equilibrium prices is an L -convex polyhedron (Theorem 11.16). This means, in particular, that p ∨ q, p ∧ q ∈ P ∗ for any p, q ∈ P ∗ and that there exist a maximum and a minimum among equilibrium prices. As a preliminary consideration, the difficulty arising from indivisible commodities is demonstrated in section 11.2 by a simple example. In section 11.3 we discuss the relevance of M -concavity as an essential property of utility functions. The results mentioned above are proved in section 11.4. Finally, in section 11.5, we show that an equilibrium can be computed by solving an M-convex submodular flow problem. Note 11.1. A special case of our economic model with L = ∅, where no producers are involved, is called the exchange economy. A difficulty of indivisible commodities already arises in this case, as we will see in section 11.2. Note 11.2. Commodities that can be represented by real-valued vectors are called divisible commodities. A framework for the rigorous mathematical treatment of equilibria in economies of divisible commodities was established around 1960 using convexity, compactness, and fixed-point theorems as major mathematical tools. See Debreu [37], [38], Nikaido [168], Arrow–Hahn [4], and McKenzie [128]. Note 11.3. A considerable literature already exists on equilibria in economies with indivisible commodities. We name a few: Henry [88], 1970; Shapley–Scarf [187], 1974; Kaneko [107], 1982; Kelso–Crawford [111], 1982; Gale [72], 1984; Quinzii [173], 1984; Svensson [196], 1984; Wako [209], 1984; Kaneko–Yamamoto [108], 1986; Van der Laan–Talman–Yang [204], 1997; Bikhchandani–Mamer [13], 1997; Danilov– Koshevoy–Murota [34], 1998 (also [35], 2001); Bevia–Quinzii–Silva [12], 1999; Gul– Stacchetti [84], 1999; and Yang [219], 2000.

11.2

Difficulty with Indivisibility

The difficulty in the mathematical treatment of indivisible commodities is illustrated by a simple example. We consider an exchange economy consisting of two agents (H = {1, 2}, L = ∅) dealing in two indivisible commodities (K = {1, 2}). Putting S = {(0, 0), (0, 1), (1, 0), (1, 1)} we define the utility functions Uh for h = 1, 2 in (11.6) by U1 (x) = min(2x(1) + 2x(2), x(1) + x(2) + 1)

(x = (x(1), x(2)) ∈ S),

U2 (x) = min(x(1) + 2x(2), 2x(1) + x(2))

(x = (x(1), x(2)) ∈ S),

where dom U1 = dom U2 = S (see Fig. 11.2). The demand correspondences D1 and D2 , calculated according to (11.8), are also given in Fig. 11.2. For instance, for

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328

Chapter 11. Application to Mathematical Economics x(2) 6 2

3

0

2

x(2) 6 1

3

0

1

x(1)

Values of utility U1 (x)

Values of utility U2 (x)

p(2) 6

p(2) 6 (1, 0)

(0, 0) 2

2

(1, 0)

1

(0, 0)

1

(1, 1)

(0, 1)

(1, 1)

(0, 1) -

0 0

x(1)

1

2

-

0

p(1)

Demand D1 (p)

0

1

2

p(1)

Demand D2 (p)

Figure 11.2. Exchange economy with no equilibrium for x◦ = (1, 1).

p = (p(1), p(2)) with 0 ≤ p(1) < 1 and 0 ≤ p(2) < 1, we have D1 (p) = {(1, 1)}; for p = (1, 1), we have D1 (p) = {(1, 1), (0, 1), (1, 0)} and D2 (p) = {(1, 1)}. Given a total initial endowment x◦ , an equilibrium is a tuple (x1 , x2 , p) ∈ Z2 × Z2 × R2+ such that x1 ∈ D1 (p),

x2 ∈ D2 (p),

x1 + x2 = x◦ .

For x◦ = (1, 2), for example, the tuple of x1 = (0, 1), x2 = (1, 1), p = (2, 1) satisfies the above conditions and hence is an equilibrium. Another case, x◦ = (1, 1), is problematic. As we have seen in (11.15), a nonnegative vector p is an equilibrium price if and only if x◦ ∈ D1 (p) + D2 (p). Superposition of the diagrams for D1 (p) and D2 (p) in Fig. 11.2 yields a similar diagram for the Minkowski sum D1 (p) + D2 (p), shown in Fig. 11.3. We see from this diagram that no p satisfies (1, 1) ∈ D1 (p) + D2 (p) and hence no equilibrium exists for x◦ = (1, 1). The diagram consists of eight regions, corresponding to the eight points in [0, 2]Z × [0, 2]Z except {(1, 1)}. Hence an equilibrium exists for every

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11.2. Difficulty with Indivisibility

329

p(2) 6 (2, 0)

(1, 0)

(0, 0)

2 (2, 1)

(0, 1)

1 (2, 2)

(1, 2)

(0, 2) -

0 0

1

2

p(1)

Figure 11.3. Minkowski sum D1 (p) + D2 (p).

x◦ ∈ ([0, 2]Z × [0, 2]Z ) \ {(1, 1)} and not for x◦ = (1, 1). Let us have a closer look at the problematic case to better understand the discreteness inherent in the problem and to identify the source of the difficulty. In view of the established mathematical framework for divisible commodities, we consider an embedding of our discrete problem via a concave extension of ˆ2 the concave extensions of U1 and U2 , ˆ1 and U the utility functions. Denote by U ˆ ˆ2 = S, where S = [0, 1]R × [0, 1]R , and respectively. Obviously, domR U1 = domR U ˆ1 (x) = min(2x(1) + 2x(2), x(1) + x(2) + 1) U ˆ U2 (x) = min(x(1) + 2x(2), 2x(1) + x(2))

(x = (x(1), x(2)) ∈ S), (x = (x(1), x(2)) ∈ S).

The demand correspondences are defined by ˆ h (p) = arg max (U ˆh (x) − p, x ) D x∈RK

(p ∈ RK )

for h = 1, 2 and an equilibrium is a tuple (x1 , x2 , p) ∈ R2 × R2 × R2+ such that ˆ 1 (p), x1 ∈ D

ˆ 2 (p), x2 ∈ D

x1 + x2 = x◦ .

In our case of x◦ = (1, 1), the tuple of x1 = x2 = (1/2, 1/2) and p = (3/2, 3/2) is an equilibrium in this sense, but it is not qualified as an equilibrium in the original problem of indivisible commodities, in which x1 and x2 must be integer vectors. Thus, there is an essential discrepancy between the original discrete problem and the derived continuous problem. We can identify the reason for this discrepancy as the lack of convexity in ˆ h (p) coincides with the convex Minkowski sum discussed in section 3.3. Since D hull Dh (p) of Dh (p) and D1 (p) + D2 (p) = D1 (p) + D2 (p) holds by Proposition 3.17 (4), the derived continuous problem has an equilibrium if and only if x◦ ∈

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Chapter 11. Application to Mathematical Economics

D1 (p) + D2 (p). On the other hand, the original discrete problem has an equilibrium if and only if x◦ ∈ D1 (p) + D2 (p), as noted already. For p = (3/2, 3/2), we have D1 (p) = {(0, 1), (1, 0)}, D2 (p) = {(0, 0), (1, 1)}, and D1 (p) + D2 (p) = {(0, 1), (1, 0), (1, 2), (2, 1)}, which has a hole at (1, 1) (see Example 3.15 and Fig. 3.4). This hole is the very reason for the nonexistence of an equilibrium for indivisible commodities.

11.3

M -Concave Utility Functions

We demonstrate the relevance of M -concavity to utility functions by indicating its relationship with fundamental properties such as submodularity, the gross substitutes property, and the single improvement property, discussed in the literature of mathematical economics. First, recall from Theorem 6.2 that we can define an M -concave function as a function U : ZK → R ∪ {−∞} with dom U = ∅ satisfying the following exchange property: (−M -EXC[Z]) For x, y ∈ dom U and i ∈ supp+ (x − y),  U (x) + U (y) ≤ max U (x − χi ) + U (y + χi ),  max

{U (x − χi + χj ) + U (y + χi − χj )} , (11.17)

j∈supp− (x−y)

where χi is the ith unit vector and a maximum taken over an empty set is defined to be −∞. A more compact expression of this exchange property is min

max

[ΔU (x; j, i) + ΔU (y; i, j)] ≥ 0,

i∈supp+ (x−y) j∈supp− (x−y)∪{0}

(11.18)

where χ0 is the zero vector, ΔU (x; j, i) = U (x − χi + χj ) − U (x) as in (6.2), ΔU (x; 0, i) = U (x − χi ) − U (x), and ΔU (y; i, 0) = U (y + χi ) − U (y). All the results established in the previous chapters for M -convex functions can obviously be rephrased for M -concave functions. In particular, an M -concave function U has the following properties (reformulations of Theorems 6.42, 6.19, 6.26, and 6.24, Propositions 6.33 and 6.35, and Theorem 6.30). ˆ of U satisfies • Concave extensibility: The concave closure U ˆ (x) = U (x) U

(x ∈ ZK ).

• Submodularity: U (x) + U (y) ≥ U (x ∨ y) + U (x ∧ y)

(x, y ∈ ZK ).

(11.19)

Utility functions are usually assumed to have decreasing marginal returns, a property that corresponds to submodularity in the discrete case.

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11.3. M -Concave Utility Functions

331

• Local characterization of global maximality: For x ∈ dom U ,  U (x) ≥ U (x − χi + χj ) (∀ i, j ∈ K), U (x) ≥ U (y) (∀ y ∈ ZK ) ⇐⇒ U (x) ≥ U (x ± χj ) (∀ j ∈ K). • (−M -SI[Z]): For p ∈ RK and x, y ∈ ZK with −∞ < U [−p](x) < U [−p](y), U [−p](x)
−∞, and u(i, j) > −∞ for any i, j ∈ K (i = j). Theorem 11.21. The set P ∗ (x◦ ) of all equilibrium price vectors is an L -convex polyhedron described as  A max{0, (j)} ≤ p(j) ≤ u(j) (j ∈ K), (11.43) P ∗ (x◦ ) = p ∈ RK p(j) − p(i) ≤ u(i, j) (i, j ∈ K, i = j) with (j), u(j), and u(i, j) defined in (11.40), (11.41), and (11.42). By Theorem 11.21, the nonemptiness of P ∗ (x◦ ) can be checked by linear programming. In particular, the largest equilibrium price vector, if any, can be

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344

Chapter 11. Application to Mathematical Economics

found by solving a linear programming problem:  p(k) Maximize k∈K

subject to

max{0, (j)} ≤ p(j) ≤ u(j) p(j) − p(i) ≤ u(i, j)

(j ∈ K), (i, j ∈ K, i = j).

(11.44)

Similarly, the smallest equilibrium price vector can be found by solving another linear programming problem:  p(k) Minimize k∈K

subject to

max{0, (j)} ≤ p(j) ≤ u(j) p(j) − p(i) ≤ u(i, j)

(j ∈ K), (i, j ∈ K, i = j).

(11.45)

Both (11.44) and (11.45) can be easily reduced to the dual of a single-source shortest path problem. Theorem 11.22. There exists an equilibrium price vector if and only if the problem (11.44) is feasible. The smallest and the largest equilibrium price vectors, if any, can be found by solving the shortest path problem. Thus, the existence of a competitive equilibrium in our economic model with M -convex cost functions of producers and M -concave utility functions of consumers can be checked in polynomial time by the following algorithm. Algorithm for computing an equilibrium S0: Construct the instance of the MSFP2 . S1: Solve the MSFP2 to obtain ((x∗h | h ∈ H), (yl∗ | l ∈ L), p). (If MSFP2 is infeasible, no equilibrium exists.) S2: Solve the problem (11.44) to obtain an equilibrium ((x∗h | h ∈ H), (yl∗ | l ∈ L), p∗ ) with largest p∗ . (If (11.44) is infeasible, no equilibrium exists.) Whereas the above algorithm yields the largest equilibrium price vector, the smallest price vector can be computed by solving (11.45) instead of (11.44) in step S2.

Bibliographical Notes The unified framework for indivisible commodities by means of discrete convex analysis is proposed in Danilov–Koshevoy–Murota [34], [35], to which Theorems 11.14 and 11.15 as well as Notes 11.9 and 11.17 are ascribed. Theorem 11.16 for the structure of equilibrium prices and Note 11.18 are by Murota [147]. The gross substitutes property was introduced by Kelso–Crawford [111] and investigated thoroughly by Gul–Stacchetti [84], in which the equivalence of (GS), (SI), and (NC) is proved. The connection of these conditions to M -concavity was pointed out by Fujishige–Yang [69] for set functions, with subsequent generalizations by Danilov–Koshevoy–Lang [33] (Theorem 11.6) and Murota–Tamura [160] (Theorems 11.4 and 11.5). See Roth–Sotomayor [180] for more on (GS).

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11.5. Computation of Equilibria

345

The computation of an equilibrium via an M-convex submodular flow problem described in section 11.5 is due to Murota–Tamura [161]. M-convexity is also amenable to the stable marriage problem (stable matching problem) of Gale–Shapley [74], which is one of the most applicable models in economics and game theory. Eguchi–Fujishige [47] formulates a generalization of the stable marriage problem in terms of M -convex functions and presents an extension of the Gale–Shapley algorithm. Submodularity plays important roles in economics and game theory. We mention here the paper of Shapley [186] as an early contribution and Bilbao [14], Danilov–Koshevoy [31], Milgrom–Shannon [129], and Topkis [203] as recent literature.

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Chapter 12

Application to Systems Analysis by Mixed Matrices This chapter presents an application of discrete convex analysis to systems analysis by mixed matrices. Motivated by a physical observation to distinguish two kinds of numbers appearing in descriptions of physical/engineering systems, the concepts of mixed matrices and mixed polynomial matrices are introduced as mathematical tools for dealing with two kinds of numbers in systems analysis. Discrete convex functions arise naturally in this context and the discrete duality theorems are vital for the analysis of the rank of mixed matrices and the degree of determinants of mixed polynomial matrices.

12.1

Two Kinds of Numbers

A physical/engineering system can be characterized by a set of relations among various kinds of numbers representing physical quantities, parameter values, incidence relations, etc., where it is important to recognize the difference in the nature of the quantities involved in the problem and to establish a mathematical model that reflects the difference. A primitive, yet fruitful, way of classifying numbers is to distinguish nonvanishing elements from zeros. This dichotomy often leads to graph-theoretic methods for systems analysis, where the existence of nonvanishing numbers is represented by a set of arcs in a certain graph. Closer inspection reveals, however, that two different kinds can be distinguished among the nonvanishing numbers; some of the nonvanishing numbers are accurate in value and others are inaccurate in value but independent of one another. We may alternatively refer to the numbers of the first kind as fixed constants and to those of the second kind as system parameters. Accurate numbers (fixed constants): Numbers accounting for various sorts of conservation laws, such as Kirchhoff’s laws, which, stemming from the topological incidence relation, are precise in value (often ±1). Inaccurate numbers (system parameters): Numbers representing independent 347

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Chapter 12. Application to Systems Analysis by Mixed Matrices

b 6

ξ1 

η1

-

η2

r1

d

r2

6

ξ5 ? η5

e

η4

ξ c -2

?ξ4 = βξ2

6

?ξ3 η3 = αη1

6

a Figure 12.1. Electrical network with mutual couplings.

physical parameters, such as resistances in electrical networks and masses in mechanical systems, which, being contaminated with noise and other errors, take values independent of one another. It is emphasized that the distinction between accurate and inaccurate numbers is not a matter in mathematics but in mathematical modeling, i.e., the way in which we recognize the problem. This means in particular that it is impossible in principle to give a mathematical definition to the distinction between the two kinds of numbers. The objective of this section is to explain, by means of typical examples, what is meant by accurate and inaccurate numbers and how numbers of different nature arise in mathematical descriptions of physical/engineering systems. We consider three examples from different disciplines: an electrical network, a chemical process, and a mechanical system. Example 12.1. Consider the electrical network in Fig. 12.1, which consists of five elements: two resistors of resistances ri (branch i) (i = 1, 2), a voltage source (branch 3) controlled by the voltage across branch 1, a current source (branch 4) controlled by the current in branch 2, and an independent voltage source of voltage e (branch 5). The element characteristics are represented as

η1 = r1 ξ1 ,

η2 = r2 ξ2 ,

η3 = αη1 ,

ξ4 = βξ2 ,

η5 = −e,

where ξi and ηi are the current in and the voltage across branch i (i = 1, . . . , 5) in the directions indicated in Fig. 12.1. We then obtain the following system of equations:

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12.1. Two Kinds of Numbers ⎡

0 1 0

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ r1 ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 0 1

1 0 −1

1 0 0

349 ⎤⎡

1 −1 0 1 0 −1

r2 α

0 β

−1 0

0 1 −1

0 1 −1

⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎢ −1 1 ⎥ ⎥⎢ ⎥ −1 0 ⎥⎢ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣ 0 −1

ξ1 ξ2 ξ3 ξ4 ξ5 η1 η2 η3 η4 η5





⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

0 0 0 0 0 0 0 0 0 e

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(12.1)

The upper five equations are the structural equations (Kirchhoff’s laws), while the remaining five are the constitutive equations. The nonzero coefficients, ±1, appearing in the structural equations represent the incidence relation in the underlying graph and are certainly accurate in value. The entries of −1 contained in the constitutive equations are also accurate by definition. In contrast, the values of the physical parameters r1 , r2 , α, and β are likely to be inaccurate, being only approximately equal to their nominal values on account of various kinds of noises and errors. The unique solvability of this network amounts to the nonsingularity of the coefficient matrix of (12.1). A direct calculation shows that the determinant of this matrix is equal to r2 + (1 − α)(1 + β)r1 , which is highly probably distinct from zero by the independence of the physical parameters {r1 , r2 , α, β}. Thus, the electrical network of this example is solvable in general or, more precisely, solvable generically with respect to the parameter set {r1 , r2 , α, β}. The solvability of this system will be treated in Example 12.11 by a systematic combinatorial method (without direct computation of the determinant). The second example concerns a chemical process simulation. Example 12.2. Consider a hypothetical system (Fig. 12.2) for the production of ethylene dichloride (C2 H4 Cl2 ), which is slightly modified from an example used in the Users’ Manual of Generalized Interrelated Flow Simulation of The Service Bureau Company. Feeds to the system are 100 mol/h of pure chlorine (Cl2 ) (stream 1) and 100 mol/h of pure ethylene (C2 H4 ) (stream 2). In the reactor, 90% of the input ethylene is converted into ethylene dichloride according to the reaction formula C2 H4 + Cl2 → C2 H4 Cl2 .

(12.2)

At the purification stage, the product ethylene dichloride is recovered and the unreacted chlorine and ethylene are separated for recycling. The degree of purification is described in terms of the component recovery ratios a1 , a2 , and a3 of chlorine, ethylene, and ethylene dichloride, respectively, which indicate the ratios of the amounts recovered in stream 6 of the respective components over those in stream 5. We consider the following problem:

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Chapter 12. Application to Systems Analysis by Mixed Matrices

6

-

recycle

31 chlorine feed 100 mol Cl2 /h 2 ethylene feed

100 mol C2 H4 /h

4 -

reactor

5 purification

C2 H4 + Cl2 −→ C2 H4 Cl2 90 % conversion of C2 H4

7 product

Figure 12.2. Hypothetical ethylene dichloride production system. Given the component recovery ratios a1 and a2 of chlorine and ethylene, determine the recovery ratio x = a3 of ethylene dichloride with which a specified production rate y mol/h of ethylene dichloride is realized. Let ui1 , ui2 , and ui3 mol/h be the component flow rates of chlorine, ethylene, and ethylene dichloride in stream i, respectively. The system of equations to be solved may be put in the following form, where u is an auxiliary variable in the reactor and r (= 0.90) is the conversion ratio of ethylene: str3 = str1 + str6: u31 = u61 + 100, u3j = u6j (j = 2, 3); str4 = str2 + str3: u42 = u32 + 100, u4j = u3j (j = 1, 3); reactor: u = ru42 , (j = 1, 2), u5j = u4j − u u53 = u43 + u, purification: u6j = aj u5j (j = 1, 2), u63 = xu53 , u7j = u5j − u6j (j = 1, 2), y = u53 − u63 .

(12.3)

This is a system of linear/nonlinear equations in the unknown variables x, u, and uij , where the equation u63 = x u53 in the purification is the only nonlinear equation. We may regard aj (j = 1, 2) and r (= 0.90) as inaccurate and independent numbers. The stoichiometric coefficients in the reaction formula (12.2) are accurate numbers. The Jacobian matrix of (12.3), shown in Fig. 12.3, contains five inaccurate numbers, a1 , a2 , r, x, and u53 . The solvability of this problem will be treated in Example 12.12.

Example 12.3. Consider the mechanical system in Fig. 12.4 consisting of two masses m1 and m2 , two springs k1 and k2 , and a damper f ; u is the force exerted

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351

x u31 u32 u33 u41 u42 u43 u51 u52 u53 u61 u62 u63 u71 u72 u y 1 −1 u31 −1 1 u32 −1 1 u33 −1 1 u41 1 −1 u42 1 −1 u43 1 −1 u51 1 −1 −1 u52 1 −1 −1 u53 1 −1 1 u61 a1 −1 u62 a2 −1 u63 u53 x −1 u71 1 −1 −1 u72 1 −1 −1 u r −1

Figure 12.3. Jacobian matrix in the chemical process simulation. from outside. This system may be described by vectors x = (x1 , x2 , x3 , x4 , x5 , x6 ) and u = (u), where x1 and x2 are the vertical displacements (downward) of masses m1 and m2 , x3 and x4 are their velocities, x5 is the force by the damper f , and x6 is the relative velocity of the two masses. The governing equation is dx ¯ u, = A¯ x + B F¯ dt with



⎢ ⎢ ⎢ F¯ = ⎢ ⎢ ⎣

1 0 0 0 0 1

0 1 0 0 0 −1

0 0 m1 0 0 0

0 0 0 m2 0 0

0 0 0 0 0 0

0 0 0 0 0 0





⎥ ⎢ ⎥ ⎢ ⎥ ¯ ⎢ ⎥, A = ⎢ ⎥ ⎢ ⎦ ⎣

0 0 −k1 0 0 0

0 0 0 −k2 0 0

(12.4)

1 0 0 0 0 0

0 1 0 0 0 0

0 0 −1 1 −1 0

0 0 0 0 f 1





⎥ ⎢ ⎥ ⎢ ⎥ ¯ ⎢ ⎥, B = ⎢ ⎥ ⎢ ⎦ ⎣

0 0 1 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦

We may regard {m1 , m2 , k1 , k2 , f } as independent system parameters and other ¯ and B ¯ as fixed constants. The Laplace nonvanishing entries (i.e., ±1) of F¯ , A, transform 67 of the equation (12.4) gives a frequency domain description   . x ¯ A¯ − sF¯ B = 0, (12.5) u ¯ is where x(0) = 0 and u(0) = 0 are assumed. The coefficient matrix [A¯ − sF¯ | B] a polynomial matrix in s with coefficients depending on the system parameters. 67 See

Chen [23] or Zadeh–Desoer [220] for the Laplace transform.

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Chapter 12. Application to Systems Analysis by Mixed Matrices

u m1 f k1

m2

x1 . x3 = x1 x2 . x4 = x 2

k2

Figure 12.4. Mechanical system.

We have employed a six-dimensional vector x in our description of the system. It is possible, however, to describe this system using a four-dimensional state vector. The minimum dimension of the state vector is known to be equal to the degree in s of the determinant of ⎤ ⎡ −s 0 1 0 0 0 ⎢ 0 −s 0 1 0 0 ⎥ ⎥ ⎢ ⎢ −k1 0 −sm1 0 −1 0 ⎥ ⎥. (12.6) A(s) = A¯ − sF¯ = ⎢ ⎢ 0 −k2 0 −sm2 1 0 ⎥ ⎥ ⎢ ⎣ 0 0 0 0 −1 f ⎦ −s s 0 0 0 1 Thus the number deg det[A¯ − sF¯ ] is an important characteristic, sometimes called the dynamical degree, of the system (12.4). As illustrated by the examples above, accurate numbers often appear in equations for conservation laws such as Kirchhoff’s laws; the law of conservation of mass, energy, or momentum; and the principle of action and reaction, where the nonvanishing coefficients are either 1 or −1, representing the underlying topological incidence relations. Another typical example is integer coefficients (stoichiometric coefficients) in chemical reactions such as 2 · H2 O = 2 · H2 + 1 · O2 , where nonunit integers such as 2 appear. In dealing with dynamical systems, we encounter another example of accurate numbers that represent the defining relations, such as those between velocity v and position x and between current ξ and charge Q: v =1·

dx , dt

ξ =1·

dQ . dt

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ξ1 j

-

* ξ2

353

−1 · ξ1 − 1 · ξ2 + 1 · ξ3 = 0

ξ3

Kirchhoff’s current law

η1



U

η3

−1 · η1 − 1 · η2 + 1 · η3 = 0

η2 Kirchhoff’s voltage law

H2 O

- H 2 - O

-

2 · H2 O = 2 · H2 + 1 · O2

2

stoichiometry

velocity v – displacement x

v = 1 · dx/dt (= s · x)

current ξ – charge Q

ξ = 1 · dQ/dt (= s · Q)

Figure 12.5. Accurate numbers. Typical accurate numbers are illustrated in Fig. 12.5. The rather intuitive concept of two kinds of numbers will be given a mathematical formalism in the next section.

12.2

Mixed Matrices and Mixed Polynomial Matrices

The distinction of two kinds of numbers can be embodied in the concepts of mixed matrices and mixed polynomial matrices. Assume that we are given a pair of fields F and K, where K is a subfield of F . Typically, K is the field Q of rational numbers and F is a field large enough to contain all the numbers appearing in the problem in question. In so doing we intend to model accurate numbers as numbers belonging to K and inaccurate numbers

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Chapter 12. Application to Systems Analysis by Mixed Matrices

as numbers in F that are algebraically independent over K, where a family of numbers t1 , . . . , tm of F is called algebraically independent over K if there exists no nonzero polynomial p(X1 , . . . , Xm ) over K such that p(t1 , . . . , tm ) = 0. Informally, algebraically independent numbers are tantamount to free parameters. A matrix A = (Aij ) over F , i.e., Aij ∈ F , is called a mixed matrix with respect to (K, F ) if A = Q + T, (12.7) where (M-Q) Q = (Qij ) is a matrix over K and (M-T) T = (Tij ) is a matrix over F such that the set of its nonzero entries is algebraically independent over K. We usually assume Tij = 0

=⇒

Qij = 0

to make the decomposition (12.7) unique. Example 12.4. In the electrical network of Example 12.1 it is reasonable to regard {r1 , r2 , α, β} as independent free parameters. Then the coefficient matrix in (12.1) is a mixed matrix with respect to (K, F ) = (Q, Q(r1 , r2 , α, β)), where Q(r1 , r2 , α, β) means the field of rational functions in r1 , r2 , α, β with coefficients from Q. The decomposition A = Q + T is given by ⎡ ⎤ 0

⎢ 1 ⎢ 0 ⎢ ⎢ ⎢ ⎢ Q=⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

0 ⎢ 0 ⎢ 0 ⎢

⎢ ⎢ ⎢ T =⎢ ⎢ r1 ⎢ ⎢ ⎢ ⎢ ⎣

0 0 1

1 0 −1

1 0 0

1 −1 0

1 0 −1 0

0 1 −1

0

0 −1

0

0 1

−1 −1

−1 0

−1

0 0 0 0

0 0 0

0 0 0

0 0 0 0 0 0

r2 β

0 0

0 0

0 0

0 α

0

⎥ ⎥ ⎥ 1 ⎥ ⎥ 0 ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0 0

0



⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 0

0

Consider now a polynomial matrix in s with coefficients from F : A(s) =

N  k=0

sk Ak ,

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355

where Ak (k = 0, 1, . . . , N ) are matrices over F . We say that A(s) is a mixed polynomial matrix with respect to (K, F ) if it can be represented as A(s) = Q(s) + T (s)

(12.8)

with Q(s) =

N 

sk Q k ,

T (s) =

k=0

N 

sk T k ,

k=0

where (MP-Q) Qk (k = 0, 1, . . . , N ) are matrices over K and (MP-T) Tk (k = 0, 1, . . . , N ) are matrices over F such that the set of their nonzero entries is algebraically independent over K. Obviously, the coefficient matrices Ak = Qk + Tk

(k = 0, 1, . . . , N )

are mixed matrices with respect to (K, F ). Also note that A(s) is a mixed matrix with respect to (K(s), F (s)) in spite of the occurrence of the variable s in both Q(s) and T (s), where K(s) and F (s) denote the fields of rational functions in variable s with coefficients from K and F , respectively. Mixed polynomial matrices are useful in dealing with linear time-invariant dynamical systems. The variable s here is primarily intended to denote the variable for the Laplace transform for continuous-time systems, though it could be interpreted as the variable for the z-transform 68 for discrete-time systems. Example 12.5. In the mechanical system of Example 12.3 it is reasonable to regard {m1 , m2 , k1 , k2 , f } as independent free parameters. Then the matrix A(s) in (12.6) is a mixed polynomial matrix with respect to (K, F ) = (Q, Q(m1 , m2 , k1 , k2 , f )). The decomposition A(s) = Q(s) + T (s) is given by ⎡ ⎤ ⎡ ⎤ ⎢ ⎢ ⎢ Q(s) = ⎢ ⎢ ⎣

−s 0 0 0 0 −s

0 −s 0 0 0 s

1 0 0 0 0 0

0 1 0 0 0 0

0 0 −1 1 −1 0

0 0 0 0 0 1

0

⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ −k1 ⎥ , T (s) = ⎢ 0 ⎥ ⎢ ⎦ ⎣ 0 0

0 0 0 −k2 0 0

0 0 −sm1 0 0 0

0 0 0 −sm2 0 0

0 0 0 0 0 0

0 0 0 0 f 0

⎥ ⎥ ⎥ ⎥. ⎥ ⎦

Our intention in the splitting (12.7) or (12.8) is to extract a meaningful combinatorial structure from the matrix A or A(s) by treating the Q-part numerically and the T -part symbolically. This is based on the following observations. 68 See

Chen [23] or Zadeh–Desoer [220] for the z-transform.

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Chapter 12. Application to Systems Analysis by Mixed Matrices

Q-part: As is typical with electrical networks, the Q-part primarily represents the interconnection of the elements. The Q-matrix, however, is not uniquely determined, but is subject to our choice in the mathematical description. In the electrical network of Example 12.1, for instance, the coefficient matrix   1 0 0 −1 1 0 1 1 −1 0 for Kirchhoff’s voltage law may well be replaced with   1 0 0 −1 1 . 1 −1 −1 0 1 Accordingly, the structure of the Q-part should be treated numerically or linear algebraically. In fact, this is feasible in practice, since the entries of the Q-matrix are usually small integers, causing no serious numerical difficulty in arithmetic operations. T -part: The T -part primarily represents the element characteristics. The nonzero pattern of the T -matrix is relatively stable against our choice in the mathematical description of constitutive equations, and therefore it can be regarded as representing some intrinsic combinatorial structure of the system. It can be treated properly by graph-theoretic concepts and algorithms. Combination: The structural information from the Q-part and the T -part can be combined properly and efficiently by virtue of the fact that each part defines a combinatorial structure with discrete convexity (matroid or valuated matroid, to be more specific). Mathematical and algorithmic results in discrete convex analysis (matroid theory and valuated matroid theory) afford effective methods of systems analysis. We may summarize the above as follows: Q-part by linear algebra T -part by graph theory Combination by matroid theory

12.3

Rank of Mixed Matrices

The rank of a mixed matrix A = Q + T can be expressed in terms of the L -convex functions (submodular set functions) associated with Q and T . This enables us, for example, to test efficiently for the solvability of the electrical network in Example 12.1 and of the chemical process simulation problem in Example 12.2. Let A = Q + T be a mixed matrix with respect to (K, F ). The rank of A is defined with reference to the field F . That is, the rank of A is equal to (i) the maximum number of linearly independent column vectors of A with coefficients taken from F , (ii) the maximum number of linearly independent row vectors of A with coefficients taken from F , and (iii) the maximum size of a submatrix of A for

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357

which the determinant does not vanish in F . The row set and the column set of A are denoted by R and C, respectively. For I ⊆ R and J ⊆ C, the submatrix of A with row indices in I and column indices in J is designated by A[I, J]. We start with the nonsingularity of a mixed matrix. Proposition 12.6. A square mixed matrix A = Q + T is nonsingular if and only if there exist some I ⊆ R and J ⊆ C such that both Q[I, J] and T [R \ I, C \ J] are nonsingular. Proof. It follows from the defining expansion of the determinant that  ε(I, J) · det Q[I, J] · det T [R \ I, C \ J], det A = |I|=|J|

with ε(I, J) ∈ {1, −1}. If A is nonsingular, we have det A = 0 and hence det Q[I, J]· det T [R \ I, C \ J] = 0 for some I and J. The converse is also true, since no cancellation occurs among nonzero terms on the right-hand side by virtue of the algebraic independence of the nonzero entries of T . The following is a basic rank identity for a mixed matrix. Theorem 12.7. For a mixed matrix A = Q + T , rank A = max{rank Q[I, J] + rank T [R \ I, C \ J] | I ⊆ R, J ⊆ C}.

(12.9)

Proof. Proposition 12.6 applied to submatrices of A establishes (12.9). The right-hand side of the identity (12.9) is a maximization over all pairs (I, J), the number of which is as large as 2|R|+|C|, too large for an exhaustive search for maximization. Fortunately, however, it is possible to design an efficient algorithm to compute this maximum on the basis of the following facts: • The function ρ(I, J) = rank Q[I, J] can be evaluated easily by Gaussian elimination. • The function τ (I, J) = rank T [I, J] can be evaluated easily by finding a maximum matching in a bipartite graph representing the nonzero pattern of T . • The maximization can be converted, with the aid of Edmonds’s intersection theorem for matroids (a special case of Theorem 4.18), to the minimum of an L -convex function. To state the main theorem of this section we need another function γ : 2R × 2 → Z defined by C

γ(I, J) = |{i ∈ I | ∃ j ∈ J : Tij = 0}|

(I ⊆ R, J ⊆ C).

Note that γ(I, J) represents the number of nonzero rows of the submatrix T [I, J].

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Chapter 12. Application to Systems Analysis by Mixed Matrices

Theorem 12.8. For a mixed matrix A = Q + T , rank A = rank A = rank A =

min {ρ(I, J) + τ (I, J) − |I| − |J|} + |R| + |C|,

(12.10)

min {ρ(I, J) + γ(I, J) − |I| − |J|} + |R| + |C|,

(12.11)

min {ρ(I, J) − |I| − |J| | γ(I, J) = 0} + |R| + |C|.

(12.12)

I⊆R,J⊆C I⊆R,J⊆C I⊆R,J⊆C

Proof. (12.10) can be proved from (12.9) with the aid of Edmonds’s intersection theorem for matroids (a special case of Theorem 4.18) and (12.11) can be derived from (12.10) using the formula τ (I, J) = min{γ(I, J  ) − |J  | | J  ⊆ J} + |J|, which is a version of the fundamental min-max relation between maximum matchings and minimum covers. (12.12) follows easily from (12.11). For details see the proofs of Theorem 4.2.11 and Corollary 4.2.12 of Murota [146]. We mention the following theorem as an immediate corollary of the third identity (12.12). Note the duality nature of this theorem. Theorem 12.9 (K˝onig–Egerv´ ary theorem for mixed matrices). For a mixed matrix A = Q + T , there exist I ⊆ R and J ⊆ C such that (i) |I| + |J| − rank Q[I, J] = |R| + |C| − rank A, and (ii) rank T [I, J] = 0. Proof. Take (I, J) that attains the minimum in (12.12). To see the connection of the above rank formulas to L -convexity, we define three functions gρ , gτ , gγ : ZR∪C → Z ∪ {+∞}, with dom gρ = dom gτ = dom gγ = {0, 1}R∪C , by gρ (χI∪J ) = ρ(R \ I, J) + |I|

(I ⊆ R, J ⊆ C),

gτ (χI∪J ) = τ (R \ I, J) + |I| gγ (χI∪J ) = γ(R \ I, J) + |I|

(I ⊆ R, J ⊆ C), (I ⊆ R, J ⊆ C).

Proposition 12.10. gρ , gτ , and gγ are L -convex functions. Proof. It is easy to see that ρ˜(I ∪ J) = ρ(R \ I, J) + |I| is a submodular set function on R ∪ C (see (2.70)). This is equivalent to the L -convexity of gρ by Theorem 7.1, and similarly for gτ and gγ . We can rewrite the right-hand sides of (12.10) and (12.11) using these L convex functions. Namely, we see (12.10) ⇐⇒ rank A = min{gρ (p) + gτ (p) − p, 1 } + |C| = |C| − (gρ + gτ )• (1), p

(12.11) ⇐⇒ rank A = min{gρ (p) + gγ (p) − p, 1 } + |C| = |C| − (gρ + gγ )• (1), p

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359

where 1 ∈ ZR∪C . Note that both gρ + gτ and gρ + gγ are L -convex and therefore (gρ + gτ )• and (gρ + gγ )• are M -convex. As for (12.12) we observe that DT = {p ∈ {0, 1}R∪C | p = χI∪J , γ(R \ I, J) = 0} is an L -convex set and (12.12) ⇐⇒ rank A = min{gρ (p) − p, χC + |C| | p ∈ DT }. This shows that the right-hand side of (12.12) is the minimum of an L -convex function over an L -convex set. The discrete convexity implicit in Theorem 12.8 is thus exposed. A concrete algorithmic procedure for computing the rank of A = Q + T is described in section 4.2 of Murota [146]. Example 12.11. The unique solvability of the electrical network in Example 12.1 can be shown by Theorem 12.7 applied to A = Q + T in Example 12.4. In (12.9) the maximum value of 10 is attained by I = {1, 2, 3, 4, 5, 7, 10} and J = {3, 4, 5, 7, 8, 9, 10}. In Theorem 12.8 the right-hand sides of (12.10), (12.11), and (12.12) are equal to 10 with the minima attained by I = R and J = ∅. Example 12.12. The generic solvability of the chemical process simulation problem in Example 12.2 is denied by Theorem 12.8 applied to the Jacobian matrix in Fig. 12.3. In (12.12) the minimum is attained by I = {y, u32 , u33 , u42 , u43 , u52 , u53 , u62 , u72 , u}, J = {x, u31 , u33 , u41 , u43 , u51 , u53 , u61 , u63 , u71 }. Note that γ(I, J) = 0, |I| = |J| = 10, and ρ(I, J) = 3, for which ρ(I, J) − |I| − |J| + |R| + |C| = 3 − 10 − 10 + 16 + 16 = 15 < |R| = |C| = 16. This shows that the Jacobian matrix in Fig. 12.3 is singular, and hence the simulation problem is not solvable in general.

12.4

Degree of Determinant of Mixed Polynomial Matrices

The degree of determinant of a mixed polynomial matrix A(s) = Q(s)+ T (s) can be expressed in terms of the infimal convolution of two M-convex functions associated with T (s) and Q(s). This enables us, for example, to compute the dynamical degree of the mechanical system in Example 12.3 in an efficient way by solving an M-convex submodular flow problem. Let A(s) = (Aij (s)) be a polynomial matrix with each entry being a polynomial in s with coefficients from a certain field F . We denote by R and C the row set and the column set of A(s). The degree of minors (subdeterminants) is an important characteristic of A(s). For example, the sequence of δk (k = 1, 2, . . .) of the highest degree in s of a minor of order k, δk = max{deg det A[I, J] | |I| = |J| = k}, I,J

(12.13)

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Chapter 12. Application to Systems Analysis by Mixed Matrices

determines the Smith–McMillan form at infinity as well as the structural indices of the Kronecker form (see section 5.1 of Murota [146]). Here the function δ(I, J) = deg det A[I, J] to be maximized in (12.13) is essentially M-concave, since ω : 2R∪C → Z ∪ {−∞} defined by ω(I ∪ J) = δ(R \ I, J) for I ⊆ R and J ⊆ C is a valuated matroid (see (2.74) and (2.77) as well as Example 5.2.15 of Murota [146]). The following is the basic identity for the degree of the determinant of a mixed polynomial matrix. Theorem 12.13. For a square mixed polynomial matrix A(s) = Q(s) + T (s), deg det A = max{deg det Q[I, J] + deg det T [R \ I, C \ J] | |I| = |J|, I ⊆ R, J ⊆ C}, (12.14) where both sides are equal to −∞ if A is singular. Proof. It follows from the defining expansion of the determinant that  ε(I, J) · det Q[I, J] · det T [R \ I, C \ J], det A = |I|=|J|

with ε(I, J) ∈ {1, −1}. Since the degree of a sum is bounded by the maximum degree of a summand, we obtain deg det A ≤ max deg(det Q[I, J] · det T [R \ I, C \ J]) |I|=|J|

= max {deg det Q[I, J] + deg det T [R \ I, C \ J]}. |I|=|J|

The inequality turns into an equality if the highest degree terms do not cancel one another. The algebraic independence of the nonzero coefficients in T (s) ensures this. The right-hand side of the identity (12.14) is a maximization over all pairs (I, J), the number of which is as large as 2|R|+|C| , too large for an exhaustive search for maximization. Fortunately, however, it is possible to compute this maximum efficiently by reducing this maximization problem to the M-convex submodular flow problem. To see the connection to M-convexity, we define functions fQ , fT : ZR∪C → Z ∪ {+∞} with dom fQ , dom fT ⊆ {0, 1}R∪C by fQ (χI∪J ) = − deg det Q[R \ I, J]

(I ⊆ R, J ⊆ C),

fT (χI∪J ) = − deg det T [R \ I, J]

(I ⊆ R, J ⊆ C).

Both fQ and fT are M-convex functions. The right-hand side of (12.14) can now be identified as the negative of an integer infimal convolution of these M-convex functions. Namely, (12.14) ⇐⇒ deg det A = −(fQ 2Z fT )(1),

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361

where 1 ∈ ZR∪C . This reveals the discrete convexity implicit in Theorem 12.13 and also shows an efficient way to compute the degree of determinant of a mixed polynomial matrix A(s) = Q(s) + T (s), since the infimal convolution of M-convex functions can be computed efficiently by solving an M-convex submodular flow problem, as we have seen in Note 9.30. Such an algorithm is described in detail in section 6.2 of Murota [146]. Example 12.14. The dynamical degree of the mechanical system in Example 12.3 can be computed by Theorem 12.13 applied to A(s) = Q(s) + T (s) in Example 12.5. In (12.14) the maximum value of 4 is attained by I = {1, 2, 5, 6} and J = {1, 2, 5, 6}. Hence the dynamical degree is equal to four.

Bibliographical Notes This chapter is largely based on Murota [146]. The observation on two kinds of numbers and the concept of mixed matrices are due to Murota–Iri [149], [150], in which Theorem 12.7 is given. Theorem 12.8 is taken from [146]. The K˝onig–Egerv´ ary theorem for mixed matrices (Theorem 12.9) is due to Bapat [7] and Hartfiel–Loewy [86]. The connection between mixed polynomial matrices and M-convexity explained in section 12.4 is due to Murota [143] and a related topic can be found in Iwata–Murota [104]. Applications of matroid theory to electrical networks are fully expounded in Iri [95] and Recski [175]. When gyrators are involved in electrical networks, a generalization of mixed matrices to mixed skew-symmetric matrices is useful, as is explained in section 7.3 of Murota [146]. See Geelen–Iwata [75] and Geelen–Iwata– Murota [76] for recent results on mixed skew-symmetric matrices. Matroid theory also finds applications in statics and scene analysis (Graver– Servatius–Servatius [80], Recski [175], Sugihara [195], Whiteley [215], [216], [217]). For planar truss structures, in particular, a necessary and sufficient condition for generic (infinitesimal) rigidity can be expressed in terms of unions of graphic matroids, where a matroid union is a special case of the Minkowski sum of two Mconvex sets. It is noted that the rigidity of a truss structure can be represented by a rank condition on a matrix associated with the truss, but that this matrix does not fall into the category of mixed matrices. Recent results on rigidity in nongeneric cases are surveyed in Radics–Recski [174].

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Index accurate number, 347 active triple, 297 acyclic, 107 admissible potential, 122 affine hull, 78 agent, 324 aggregate cost function, 335 aggregation of function to subset, 143, 162 by network transformation, 272 algebraically independent, 354 algorithm competitive equilibrium, 344 conjugate scaling, 320 cycle-canceling, 313 domain reduction, 284 domain reduction scaling, 287 fully combinatorial, 290 greedy, 3, 108 IFF fixing, 300 IFF scaling, 299 L-convex function minimization, 305, 306, 308 M-convex function minimization, 281, 283, 284, 287 primal-dual, 315 pseudopolynomial, 288 Schrijver’s, 293 steepest descent, 281, 305, 306 steepest descent scaling, 283, 308 strongly polynomial, 288 submodular function minimization, 293, 299, 300 successive shortest path, 312 two-stage, 310 weakly polynomial, 288 arc, 52

entering, 53 leaving, 52 augmenting path, 60, 273, 274 δ-, 297 auxiliary network, 252, 263 base, 105 extreme, 105 matrix, 69 matroid, 70 base family matrix, 69 matroid, 70 valuated matroid, 72 base polyhedron, 18, 105 integral, 18 biconjugate function, 82 integer, 212 bipartite graph, 89 bipartite matching, 89 Birkhoff’s representation theorem, 292 Boolean lattice, 104 boundary, 53 branch, 52 budget set, 324 certificate of optimality, 12 chain, 88 characteristic curve, 54, 251 discrete, 57 characteristic vector, 16 chemical process, 349 Choquet integral, 16, 104 closed convex function, 79 closed convex hull, 78 closed interval, 77 closure 381

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382 concave function, 216 convex function, 93 convex set, 78 coboundary, 53, 248 another convention, 253 cocontent, 55 combinatorial optimization, 3 commodity divisible, 327 indivisible, 323 compartmental matrix, 43 competitive economy, 324 competitive equilibrium, 325 complementarity, 88 complements, 62 concave closure, 216 concave conjugate, 11, 81 discrete, 212 concave extensible, 93 concave extension, 93 concave function, 9, 78 quasi-separable, 334 separable, 333 conductance, 41 cone, 78 convex, 78 L-convex, 131 M-convex, 119 polar, 82 conformal decomposition, 64 conjugacy theorem closed proper M-/L-convex, 210 in convex analysis, 11, 82 discrete M-/L-convex, 30, 212 polyhedral M-/L-convex, 209 conjugate function concave, 81, 212 convex, 81, 212 conjugate scaling, 319 conjugate scaling algorithm, 320 conservation law, 54 constitutive equation, 54, 349 constraint, 1 consumer, 323 consumption, 323 content, 55

Index contraction normal, 45 unit, 45 convex closure function, 93 set, 78 convex combination, 78 convex cone, 78 convex conjugate, 10, 81 discrete, 212 convex extensible, 93 convex extension, 93 local, 93 convex function, 2, 9, 77 closed, 79 dual-integral polyhedral, 161 integral polyhedral, 161 laminar, 141 polyhedral, 80 positively homogeneous, 82 proper, 77 quadratic, 40 quasi, 168 quasi-separable, 140 separable, 10, 95, 140, 182 strictly, 77 univariate, 10 convex hull, 78 closed, 78 convex polyhedron, 78 convex program, 2 M-, 235 convex set, 2, 78 convexity discrete midpoint, 23, 129, 180 function, 77 in intersection, 92 midpoint, 9 in Minkowski sum, 92 quasi, 168 set, 78 convolution infimal, 80 integer infimal, 143 by network transformation, 272 cost function

sidca00si 2013/2/12 page 383

Index aggregate, 335 flow, 53, 246, 255, 256 flow boundary, 256 producer’s, 324 reduced, 249 tension, 53 current, 41, 53 current potential, 55 cut capacity function, 247 cycle negative, 122, 252, 263 simple, 62 cycle-canceling algorithm, 313 decreasing marginal return, 330 demand correspondence, 325 set, 325 descent direction, 147 diagonal dominance, 41 directed graph, 52, 88 directional derivative, 80 Dirichlet form, 45 discrete Legendre–Fenchel transformation, 13, 212 discrete midpoint convexity function, 23, 180 set, 129 discrete separation theorem generic form, 13, 216 L-convex function, 218 L-convex set, 36, 126 M-convex function, 217 M-convex set, 36, 114 submodular function, 17, 111 submodular function (as special case of L-separation), 33, 224 discreteness in direction, 10 in value, 13 distance function, 122 distributive lattice, 292 distributive law, 292 divisible commodity, 327 domain reduction algorithm, 284

383 domain reduction scaling algorithm, 287 dual-integral polyhedral convex function, 161 L-convex function, 191 M-convex function, 161 dual integrality intersection theorem, 20, 114 linear programming, 89 minimum cost flow problem, 252 polyhedral convex function, 161 polyhedral L-convex function, 191 polyhedral M-convex function, 161 submodular flow problem, 261 dual linear program, 87 dual problem, 87 dual variable, 53 duality, 2, 11 Edmonds’s intersection theorem, 20 Fenchel, 85 Fenchel-type, 222, 225 L-separation, 218 linear programming, 87 M-separation, 217 matroid intersection, 225 separation for convex functions, 84 separation for convex sets, 35, 83 separation for L-convex functions, 218 separation for M-convex functions, 217 separation for submodular functions, 17, 111 strong, 87 valuated matroid intersection, 225 weak, 87 weight splitting, 34, 225 dynamical degree, 352 economy of Arrow–Debreu type, 323 Edmonds’s intersection theorem, 3, 20, 112

sidca00si 2013/2/12 page 384

384 (as special case of Fenchel-type duality), 34, 224 effective domain function over Rn , 9, 21, 77 function over Zn , 21 set function, 103 electrical network, 41, 43, 348 multiterminal, 53 elementary vector, 64 entering arc, 53 epigraph, 79 equilibrium competitive, 325 economy, 325 electrical network, 55 exchange axiom (B-EXC[R]), 118 (B-EXC+ [R]), 118 (B-EXC[Z]), 18, 101 (B-EXC+ [Z]), 102 (B-EXC− [Z]), 103 (B-EXCw [Z]), 103 (B -EXC[R]), 118 (B -EXC[Z]), 117 local, 135 M-convex function, 26, 58, 133 M-convex polyhedron, 118 M-convex set, 18, 101 M -convex function, 27, 134 M -convex polyhedron, 118 M -convex set, 117 (M-EXC[R]), 29, 56, 160 (M-EXC [R]), 160 (M-EXC[Z]), 26, 58, 133 (M-EXC [Z]), 26, 133 (M-EXCloc [Z]), 135 (M-EXCw [Z]), 137 (M -EXC[R]), 29, 47, 162 (M -EXC [R]), 162 (M -EXC+ [R]), 48 (M -EXC[Z]), 27, 134 (M -EXC [Z]), 134 (−M -EXC[Z]), 330 matroid, 69 multiple, 333

Index polyhedral M-convex function, 29, 56, 160 polyhedral M -convex function, 29, 47, 162 simultaneous, 69 weak, 137 exchange capacity, 284, 312 exchange economy, 327 extension concave, 93 convex, 93 distance function, 165 local convex, 93 Lov´ asz, 16, 104, 111 partial order, 108 set function, 16, 104 extreme base, 105 Farkas lemma, 50, 87 feasible δ-, 296 dual problem, 236 flow, 247, 258 minimum cost flow problem, 247 potential, 122 primal problem, 235 set, 1 submodular flow problem, 258 Fenchel duality, 12, 85 Fenchel transformation, 81 Fenchel-type duality generic form, 13 L-convex function, 32, 222 M-convex function, 32, 222 submodular function, 225 fixed constant, 347 flow, 53 δ-feasible, 296 feasible, 247, 258 Frank’s discrete separation theorem, 17, 111 (as special case of L-separation), 33, 224 Frank’s weight-splitting theorem, 34, 225 fully combinatorial algorithm, 290

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Index fundamental circuit, 149 g-polymatroid, 117 generalized polymatroid, 117 generator, 45 global minimizer, 79 global optimality, 2 global optimum, 9 goods divisible, 327 indivisible, 323 gradient, 80 graph acyclic, 107 bipartite, 89 directed, 52, 88 Grassmann–Pl¨ ucker relation, 69 greedy algorithm, 3, 108 gross substitutes property, 153, 331 stepwise, 155, 331 ground set, 70 gyrator, 361 Hamiltonian path problem, 257 hole free, 90 ideal, 107 IFF fixing algorithm, 300 IFF scaling algorithm, 298, 299 in kilter, 314 inaccurate number, 347 incidence chain, 88 graph, 88 topological, 347 income, 324 independent set matrix, 68 matroid, 70 indicator function, 79, 90 indivisible commodity, 323 indivisible goods, 323 infimal convolution, 80 integer, 143 by network transformation, 272 initial endowment, 324

385 total, 326 initial vertex, 53 inner product, 79 integer biconjugate, 212 integer infimal convolution, 143 integer interval, 92 integer subdifferential, 166 integral base polyhedron, 18 integral L-convex polyhedron, 131 integral M-convex polyhedron, 118 integral neighborhood, 93 integral polyhedral convex function, 161 L-convex function, 191 L -convex function, 192 M-convex function, 161 M -convex function, 162 integral polyhedron, 90 integrality dual, 161, 252, 261 linear programming, 89 minimum cost flow problem, 252 polyhedral convex function, 161 polyhedral L-convex function, 191 polyhedral M-convex function, 161 polyhedron, 90 primal, 252, 261 submodular flow problem, 261 integrally concave function, 94 integrally convex function, 7, 94 submodular, 189 integrally convex set, 96 intersection convexity in, 92 M-convex, 219 matroid, 3 submodular polyhedron, 20 valuated matroid, 225 intersection theorem Edmonds’s, 3, 20, 112 Edmonds’s (as special case of Fencheltype duality), 34, 224 M-convex, 219 valuated matroid, 225 weighted matroid, 225 interval

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386

Index closed, 77 integer, 92 open, 77

jump system, 120 kilter diagram, 53, 251 in, 314 out of, 314 Kirchhoff’s law, 349 current, 353 voltage, 353 K˝ onig–Egerv´ ary theorem for mixed matrix, 358 L-concave function, 22 polyhedral, 190 L-convex cone, 131 L-convex function, 8, 22, 177 dual-integral polyhedral, 191 integral polyhedral, 191 polyhedral, 190 positively homogeneous, 193 quadratic, 52, 182 quasi, 199 semistrictly quasi, 199 L-convex polyhedron, 123, 131 integral, 131 L-convex set, 22, 121 L-optimality criterion, 185, 193 quasi, 201 L-proximity theorem, 186 quasi, 201 L-separation theorem, 33, 218 L2 -optimality criterion, 232 L2 -proximity theorem, 232 L2 -convex function, 229 L2 -convex set, 128 L2 -convex function, 229 L2 -convex set, 129 L -convex function, 8, 23, 178 integral polyhedral, 192 polyhedral, 192 quadratic, 48, 52, 182 L -convex polyhedron, 129, 131

L -convex set, 121, 128 Lagrange duality, 234 Lagrangian function, 236 dual, 242 laminar convex function, 141 by network transformation, 273 laminar family, 141 Laplace transform, 351 lattice, 292 distributive, 292 sub-, 104 leading principal minor, 40 leading principal submatrix, 40 leaving arc, 52 Legendre–Fenchel transform concave, 11, 81 convex, 10, 81 discrete, 13, 212 Legendre–Fenchel transformation concave, 11, 81 convex, 10, 81 discrete, 13, 212 Legendre transformation, 81 level set, 172 linear extension partial order, 108 set function, 16, 104 linear order, 108 linear program, 87 dual problem, 87 primal problem, 87 linear programming, 86 duality, 87 linearity in direction 1, 177, 190 local convex extension, 93 local optimality, 2 local optimum, 10 Lov´ asz extension, 16, 104, 111 LP, 87 duality, 87 M-concave function, 8, 26 polyhedral, 160 M-convex cone, 119 M-convex function, 8, 26, 133 dual-integral polyhedral, 161

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Index integral polyhedral, 161 polyhedral, 160 positively homogeneous, 164 quadratic, 52, 139 quasi, 169 semistrictly quasi, 169 M-convex intersection problem, 219, 264 theorem, 219 M-convex polyhedron, 108, 118 integral, 118 M-convex program, 235 M-convex set, 27, 101 M-convex submodular flow problem, 256 economic equilibrium, 341 M-matrix, 42 M-minimizer cut, 149 with scaling, 158 M-optimality criterion, 148, 163 quasi, 173 M-proximity theorem, 156 quasi, 174 M-separation theorem, 33, 217 M2 -optimality criterion, 227, 228 M2 -proximity theorem, 228 M2 -convex function, 226 M2 -convex set, 116 M2 -convex function, 226 M2 -convex set, 117 M -convex function, 8, 27, 134 integral polyhedral, 162 polyhedral, 161 quadratic, 48, 52, 139 M -convex polyhedron, 117, 118 M -convex set, 102, 117 Markovian, 45 matching, 89 bipartite, 89 perfect, 89 weighted, 89, 266 mathematical programming, 1 matrix compartmental, 43 incidence (chain), 88

387 incidence (graph), 88 M-, 42 mixed, 354 mixed polynomial, 355 mixed skew-symmetric, 361 node admittance, 42 polynomial, 71, 354 positive-definite, 39 positive-semidefinite, 39 principal sub-, 40 totally unimodular, 88 matroid, 70 induction through a graph, 270 intersection problem, 34, 225 valuated, 72 max-flow min-cut theorem for submodular flow, 259 maximum submodular flow problem, 259 maximum weight circulation problem, 61 mechanical system, 350 midpoint convexity, 9 discrete function, 23, 180 discrete set, 129 Miller’s discrete convex function, 98 min-max relation, 2 minimizer, 79 global, 9, 79 integrally convex function, 94 L-convex function, 185, 305 L2 -convex function, 232 local, 10 M-convex function, 148, 281 M2 -convex function, 227, 228 maximal, 290, 291, 304, 307 minimal, 290, 291, 305, 307 submodular set function, 288 minimizer cut M-convex function, 149 M-convex function with scaling, 158 quasi M-convex function, 174 quasi M-convex function with scaling, 175 minimum cost flow problem, 53, 245

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388 integer flow, 246 minimum cut, 316 minimum spanning tree problem, 149 Minkowski sum, 80, 90 convexity in, 92 discrete, 90 integral, 90 minor, 40, 359 leading principal, 40 principal, 40 mixed matrix, 354 mixed polynomial matrix, 355 mixed skew-symmetric matrix, 361 money, 323 monotonicity, 54 multimodular function, 183 multiple exchange axiom, 333 multiterminal electrical network, 53 negative cycle, 122, 252, 263 criterion, 252, 263, 264 negative support, 18 neighborhood, integral, 93 network, 53 auxiliary, 252, 263 electrical, 53 transformation by, 270 network flow duality, 268 electrical network, 41, 43 L-convexity, 24, 31, 56, 58, 270 M-convexity, 28, 31, 56, 58, 270 maximum weight circulation, 61 minimum cost flow, 245 multiterminal, 53 submodular flow, 255 no complementarities property, 332 strong, 332 node, 52 node admittance matrix, 42 normal contraction, 45 objective function, 1 off-diagonal nonpositivity, 41 open interval, 77 optimal potential, 89, 251

Index optimal value function, 236 optimality global, 2, 9 local, 2, 10 optimality criterion integrally convex function, 94, 95 L-convex function, 185, 193 L2 -convex function, 232 M-convex function, 148, 163 M-convex submodular flow, 262– 264 M2 -convex function, 219, 227, 228 minimum cost flow, 249, 252 by negative cycle, 252, 263, 264 by potential, 249, 260, 262 quasi L-convex function, 201 quasi M-convex function, 173 submodular flow, 260 submodular set function, 185 sum of M-convex functions, 219 valuated matroid intersection, 225 weighted matroid intersection, 225 optimization combinatorial, 3 continuous, 1 discrete, 3 optimum global, 9 local, 10 out of kilter, 314 pairing, 79 parallel, 62 partial order acyclic graph, 107 extreme base, 108 perfect matching, 89 minimum weight, 89, 266 Poisson equation, 41, 43, 47 polar cone, 82 polyhedral convex function, 25, 80 L-concave function, 190 L-convex function, 190 L -convex function, 192 M-concave function, 160

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Index M-convex function, 160 M -convex function, 161 method, 8 polyhedron base, 105 convex, 78 integral, 90 integral L-convex, 131 integral M-convex, 118 L-convex, 123, 131 L -convex, 129, 131 M-convex, 108, 118 M -convex, 117, 118 rational, 90 submodular, 112 polynomial matrix, 71, 354 mixed, 355 polytope, 90 positive definite, 39 positive semidefinite, 39 positive support, 18 positively homogeneous function, 7, 82 L-convex function, 193 M-convex function, 164 potential, 41, 53, 89, 248 criterion, 249, 260, 262 optimal, 251 primal-dual algorithm, 315 primal integrality, 252, 261 intersection theorem, 20, 114 linear programming, 89 primal problem, 87 principal minor, 40 principal submatrix, 40 leading, 40 producer, 323 production, 323 profit, 324 function, 324 projection base polyhedron, 117 function to subset, 143, 162 M-convex function, 134 M-convex polyhedron, 118 M-convex set, 102

389 M2 -convex function, 226 M2 -convex set, 117 polyhedral M-convex function, 161 proper convex function, 77 proximity theorem, 156 L-convex function, 186 L2 -convex function, 232 M-convex function, 156 M2 -convex function, 228 quasi L-convex function, 201 quasi M-convex function, 174 pseudopolynomial algorithm, 288 quadratic form, 39 function, 39 L-convex function, 182 L -convex function, 48, 52, 182 M-convex function, 139 M -convex function, 48, 52, 139 quasi convex, 168 semistrictly, 168 quasi L-convex function, 199 quasi L-optimality criterion, 201 quasi L-proximity theorem, 201 quasi linear, 324 quasi M-convex function, 169 quasi M-minimizer cut, 174 with scaling, 175 quasi M-optimality criterion, 173 quasi M-proximity theorem, 174 quasi-separable concave function, 334 convex function, 140 quasi submodular, 198 semistrictly, 198 rank function matrix, 69 matroid, 70 rational polyhedron, 90 reduced cost, 249, 251 relative interior, 78 reservation value function, 325 resistance, 41 resolvent, 45

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390 resource allocation problem, 4, 176 restriction function to interval, 92 function to subset, 143, 162 L-convex function, 178 L-convex polyhedron, 131 L-convex set, 121 L2 -convex function, 229 L2 -convex set, 129 polyhedral L-convex function, 192 rigidity, 361 ring family, 104, 107, 292 saddle-point theorem, 238 scaling, 145 conjugate, 319 cost, 318 domain, 145 nonlinear, 170, 199 scaling algorithm L-convex function, 308 M-convex function, 283, 287 M-convex submodular flow, 320 semigroup, 45 semistrictly quasi convex, 168 semistrictly quasi L-convex, 199 semistrictly quasi M-convex, 169 semistrictly quasi submodular, 198 separable concave function, 333 quasi, 334 separable convex function, 10, 95, 140, 182 with chain condition, 182 quasi, 140 separation theorem convex function, 2, 11, 84 convex set, 35, 83 generic discrete, 13, 216 L-convex function, 218 L-convex set, 36, 126 M-convex function, 217 M-convex set, 36, 114 submodular function, 17, 111 series, 62 simple cycle, 62 single improvement property, 332

Index spanning tree, 149 stable marriage problem, 345 stable matching problem, 345 steepest descent algorithm L-convex function, 305, 306 M-convex function, 281 steepest descent scaling algorithm L-convex function, 308 M-convex function, 283 stepwise gross substitutes property, 155, 331 stoichiometric coefficient, 350 strictly convex function, 77 strong duality, 87 strong no complementarities property, 332 strongly polynomial algorithm, 288 structural equation, 54, 349 subdeterminant, 40, 359 subdifferential, 80 concave function, 217 discrete function, 166 integer, 166 subgradient, 80 discrete function, 166 sublattice, 104 submatrix leading principal, 40 submodular, 62, 206 function, 16, 44, 70, 104 function on distributive lattice, 292 integrally convex function, 7, 189 polyhedron, 20, 112 utility function, 330 submodular flow problem, 255 economic equilibrium, 341 feasibility theorem, 258 M-convex, 256 maximum, 259 submodular function minimization IFF fixing algorithm, 300 IFF scaling algorithm, 298, 299 Schrijver’s algorithm, 293 submodularity, 44, 177, 190 inequality, 16, 44, 177 local, 180

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Index substitutes, 62 successive shortest path algorithm, 312 sum of functions, 80 of M-convex functions, 226 supermodular, 16, 62, 105, 145, 206 function, 105 supply correspondence, 324 set, 324 support function, 82 negative, 18 positive, 18 system parameter, 347 tension, 53 another convention, 253 terminal vertex, 52, 53 tight set, 108 transformation by network, 269 of flow type, 269 of potential type, 269 transitive, 107, 119 translation submodularity, 23, 44, 178 triangle inequality, 24, 122 two-stage algorithm, 310 unimodular totally, 88 unique-min condition, 266 unit contraction, 45 unit demand preference, 334 univariate function, 10 discrete convex, 95 polyhedral convex, 80 utility function, 324 valuated matroid, 7, 72, 225 intersection problem, 225 valuation, 72 variational formulation, 43, 55 vertex, 52 initial, 53 terminal, 53 voltage, 41, 53

391 voltage potential, 55 weak duality, 87 weak exchange axiom, 137 weakly polynomial algorithm, 288 weight splitting matroid intersection, 34, 225 valuated matroid intersection, 225 z-transform, 355