133 83 5MB
English Pages [406] Year 2003
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. MConvex Sets and Submodular Set Functions
5. LConvex Sets and Distance Functions
6. MConvex Functions
7. LConvex 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
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. EditorinChief 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 MarnelaVallée, 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, BarIlan 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 LocalitySensitive 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 191042688 USA. Library of Congress CataloginginPublication Data Murota, Kazuo, 1955Discrete convex analysis / Kazuo Murota. p. cm. — (SIAM monographs on discrete mathematics and applications) Includes bibliographical references and index. ISBN 9781611972559 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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sidca00si 2013/2/12 page v
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 LConvex Functions . . . . . . . . . . 1.4.2 MConvex 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 MMatrices . . . . . . . . . . . . . . 2.1.3 Combinatorial Property of Conjugate Functions 2.1.4 General Quadratic L/MConvex Functions . . . 2.2 Nonlinear Networks . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 RealValued Flows . . . . . . . . . . . . . . . . . 2.2.2 IntegerValued Flows . . . . . . . . . . . . . . . 2.2.3 Technical Supplements . . . . . . . . . . . . . . . 2.3 Substitutes and Complements in Network Flows . . . . . . . 2.3.1 Convexity and Submodularity . . . . . . . . . .
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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
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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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MConvex Sets and Submodular Set Functions 4.1 Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Exchange Axioms . . . . . . . . . . . . . . . . . . . . 4.3 Submodular Functions and Base Polyhedra . . . . . . 4.4 Polyhedral Description of MConvex Sets . . . . . . . 4.5 Submodular Functions as Discrete Convex Functions 4.6 MConvex Sets as Discrete Convex Sets . . . . . . . . 4.7 M Convex Sets . . . . . . . . . . . . . . . . . . . . . 4.8 MConvex Polyhedra . . . . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . .
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101 101 102 103 108 111 114 116 118 119
LConvex Sets and Distance Functions 5.1 Deﬁnition . . . . . . . . . . . . . . . . . . . . 5.2 Distance Functions and Associated Polyhedra 5.3 Polyhedral Description of LConvex Sets . . . 5.4 LConvex Sets as Discrete Convex Sets . . . . 5.5 L Convex Sets . . . . . . . . . . . . . . . . . 5.6 LConvex Polyhedra . . . . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . .
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121 121 122 123 125 128 131 131
MConvex Functions 6.1 MConvex 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 MConvex Functions . . . . . . . . . 6.12 Positively Homogeneous MConvex Functions .
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6.13 Directional Derivatives and Subgradients . . . . . . . . . . . . . 166 6.14 Quasi MConvex Functions . . . . . . . . . . . . . . . . . . . . . 168 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7
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LConvex Functions 7.1 LConvex 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 LConvex Functions . . . . . . . . 7.9 Positively Homogeneous LConvex Functions . 7.10 Directional Derivatives and Subgradients . . . 7.11 Quasi LConvex 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/LConvex Functions . . . . . 8.1.3 Integral M/LConvex Functions . . . . . . . 8.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Separation Theorems . . . . . . . . . . . . . 8.2.2 FenchelType 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 MConvexity 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 MConvex 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 NegativeCycle 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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Algorithms 10.1 Minimization of MConvex 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 LConvex 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 MConvex Submodular Flows . . . . . . . . . . 10.4.1 TwoStage Algorithm . . . . . . . . . . . . . . . . 10.4.2 Successive Shortest Path Algorithm . . . . . . . . 10.4.3 CycleCanceling Algorithm . . . . . . . . . . . . . 10.4.4 PrimalDual Algorithm . . . . . . . . . . . . . . . 10.4.5 Conjugate Scaling Algorithm . . . . . . . . . . . . Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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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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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Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 Bibliography
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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 (BEXC[Z]) . . . . . . . . . . . . . . . . . . . . Deﬁnition of Lconvexity . . . . . . . . . . . . . . . . . . . . . . . Discrete midpoint convexity . . . . . . . . . . . . . . . . . . . . . . Property of a convex function . . . . . . . . . . . . . . . . . . . . . Exchange property in the deﬁnition of Mconvexity . . . . . . . . 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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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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L convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Discrete midpoint convexity . . . . . . . . . . . . . . . . . . . . . . 129 xi
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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 Quasiconvex function . . . . . . . . . . . . . . . . . . . . . . . . . 168
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Discrete midpoint convexity . . . . . . . . . . . . . . . . . . . . . . 181
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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 Mconvex intersection problem Transformation by a network . . . . . . . . . . . . . . . . . . Bipartite graphs for aggregation and convolution operations . Rooted directed tree for a laminar family . . . . . . . . . . .
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ˆ at v . . . . . . . . . . . . . . . . . . . . . . 316 Structure of G and G α Conjugate scaling f and scaling g α for α = 2 . . . . . . . . . . 320
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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 . . . . . . . . . . .
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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 . . . . . . . . . . . . . . . . . . .
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Notation Mconvex Lconvex
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: Mconvex set, Mconvex polyhedron (B): simultaneous exchange axiom of matroids
section 4.1 section 2.4
(3.16) section 2.4
sidca00si 2013/2/12 page xv
Notation
xv
B: base family of a matroid B(ρ): base polyhedron deﬁned by submodular set function ρ (BEXC[R]): exchange axiom of Mconvex polyhedra (BEXC+ [R]): exchange axiom of Mconvex polyhedra (BEXC[Z]): exchange axiom of Mconvex sets (BEXCw [Z]): exchange axiom of Mconvex sets (BEXC+ [Z]): exchange axiom of Mconvex sets (BEXC− [Z]): exchange axiom of Mconvex 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 integervalued discrete convex functions section 3.4 C[R → R]: set of univariate polyhedral convex functions section 3.1 C[ZR → R]: set of univariate integral polyhedral convex functions section 6.11 C[R → RZ]: set of univariate dualintegral polyhedral convex functions section 6.11 D: Lconvex set, Lconvex polyhedron section 5.1, section 5.6 section 11.1 Dh : demand function of consumer h D(γ): Lconvex 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, Mconvex 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: Lconvex 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, Mconcave 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: Lconcave 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 Lconvex sets L˜0 [Z]: set of indicator functions of Lconvex sets L0 [R]: set of Lconvex polyhedra L0 [ZR]: set of integral Lconvex 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 [ZR]: set of integral L convex polyhedra L[Z → R]: set of Lconvex functions L[Z → Z]: set of integervalued Lconvex functions L[R → R]: set of polyhedral Lconvex functions L[ZR → R]: set of integral polyhedral Lconvex functions L[R → RZ]: set of dualintegral polyhedral Lconvex functions L [Z → R]: set of L convex functions L [Z → Z]: set of integervalued L convex functions L [R → R]: set of polyhedral L convex functions L [ZR → R]: set of integral polyhedral L convex functions L [R → RZ]: set of dualintegral 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
sidca00si 2013/2/12 page xvii
Notation
xvii
0 L[R
→ R]: set of positively homogeneous polyhedral Lconvex functions section 7.9 L[ZR → R]: set of positively homogeneous integral polyhedral Lconvex 0 functions section 7.9 0 L[R → RZ]: set of positively homogeneous dualintegral polyhedral Lconvex functions section 8.1.2 L[Z → R]: set of positively homogeneous Lconvex functions section 7.9 0 0 L[Z → Z]: set of positively homogeneous integervalued Lconvex functions section 7.9 section 8.3 L2 [Z → R]: set of L2 convex functions section 8.3 L2 [Z → Z]: set of integervalued L2 convex functions L2 [Z → R]: set of L2 convex functions
section 8.3
L2 [Z
→ Z]: set of integervalued functions (L APR[Z]): property of L convex functions
section 8.3 section 7.2
max: maximum min: minimum M0 [Z]: set of Mconvex sets ˜ 0 [Z]: set of indicator functions of Mconvex sets M M0 [R]: set of Mconvex polyhedra M0 [ZR]: set of integral Mconvex 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 [ZR]: set of integral M convex polyhedra M[Z → R]: set of Mconvex functions section 6.1 M[Z → Z]: set of integervalued Mconvex functions section 6.1 M[R → R]: set of polyhedral Mconvex functions section 6.11 M[ZR → R]: set of integral polyhedral Mconvex functions section 6.11 M[R → RZ]: set of dualintegral polyhedral Mconvex functions section 6.11 section 6.1 M [Z → R]: set of M convex functions section 6.1 M [Z → Z]: set of integervalued M convex functions section 6.11 M [R → R]: set of polyhedral M convex functions section 6.11 M [ZR → R]: set of integral polyhedral M convex functions M [R → RZ]: set of dualintegral polyhedral M convex functions section 8.1.2 0 M[R → R]: set of positively homogeneous polyhedral Mconvex functions section 6.12 0 M[ZR → R]: set of positively homogeneous integral polyhedral Mconvex functions section 6.12 0 M[R → RZ]: set of positively homogeneous dualintegral polyhedral Mconvex functions section 8.1.2 M[Z → R]: set of positively homogeneous Mconvex functions section 6.12 0
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xviii
Notation
0 M[Z
→ Z]: set of positively homogeneous integervalued Mconvex functions section 6.12 section 8.3 M2 [Z → R]: set of M2 convex functions section 8.3 M2 [Z → Z]: set of integervalued M2 convex functions M2 [Z → R]: set of M2 convex functions M2 [Z
section 8.3
M2 convex
→ Z]: set of integervalued functions section 8.3 (MEXC[Z]): exchange axiom of Mconvex functions section 1.4.2, section 6.1 section 1.4.2, section 6.1 (MEXC [Z]): exchange axiom of Mconvex functions section 6.2 (MEXCloc [Z]): local exchange axiom of Mconvex functions (MEXCw [Z]): weak exchange axiom of Mconvex functions section 6.2 (MEXC[R]): exchange axiom of polyhedral Mconvex functions section 1.4.2, section 6.11 section 6.11 (MEXC [R]): exchange axiom of polyhedral Mconvex 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 (MGS[Z]): gross substitutes property of Mconvex 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 (MSI[Z]): descent property of Mconvex 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 : Mconvex submodular ﬂow problem (linear arc cost) MSFP3 : Mconvex 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 Lconvex function
(3.58) section 1.4.1
sidca00si 2013/2/12 page xix
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 (gpolymatroid) 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 realvalued submodular set functions S[Z]: set of integervalued 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 realvalued distance functions with triangle inequality T [Z]: set of integervalued 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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Notation
(VM): axiom of valuated matroids
section 2.4.2
x: variable of an Mconvex 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
sidca00si 2013/2/12 page xxi
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 Lconvexity and Mconvexity, play primary roles, where “L” stands for “lattice” and “M” for “matroid.” Lconvex functions and Mconvex 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. Lconvex functions and Mconvex functions generalize, respectively, the concepts of submodular set functions and base polyhedra of (poly)matroids. Lconvexity and Mconvexity prevail in discrete systems. • In network ﬂow problems, ﬂow and tension are dual objects. Roughly speaking, ﬂow corresponds to Mconvexity and tension to Lconvexity. • In matroids, the rank function corresponds to Lconvexity and the base family to Mconvexity. • Mmatrices in matrix theory correspond to Lconvexity and their inverses to Mconvexity. Hence, in a discretization of the Poisson problem of partial diﬀerential equations, for example, the diﬀerential operator corresponds to Lconvexity and the Green function to Mconvexity. • Dirichlet forms in probability theory are essentially the same as quadratic Lconvex functions. This book is intended to be read proﬁtably by graduate students in operations research, mathematics, and computer science and also by mathematicsoriented xxi
sidca00si 2013/2/12 page xxii
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Preface
practitioners and applicationoriented mathematicians. Selfcontained 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 wellbehaved 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 ﬁnitedimensional 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 ndimensional real vectors.
1
sidca00si 2013/2/12 page 2
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 minmax relation or the separation theorem, holds good. This leads, for instance, to primaldual algorithms using dual variables and also to sensitivity analysis in terms of dual variables.
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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 primaldual 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 socalled 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 minmax 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.
sidca00si 2013/2/12 page 4
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 integervalued functions. Another reformulation of Edmonds’s intersection theorem is Fujishige’s Fencheltype minmax duality theorem for a pair of submodular/supermodular functions, again with an integrality assertion in the case of integervalued 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 socalled 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
sidca00si 2013/2/12 page 5
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 Lconvexity and Mconvexity, play primary roles in the present theory. Lconvex functions and Mconvex functions are both (extensible to) convex functions and they are conjugate to each other through a discrete version of the Legendre–Fenchel transformation. Lconvex functions and Mconvex functions generalize, respectively, the concepts of submodular set functions and base polyhedra. It is noted that the “L” in “Lconvexity” stands for “lattice” and the “M” in “Mconvexity” 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 Lconvex and Mconvex 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 Lconvex and Mconvex 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 minmax 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 minmax 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 ndimensional integer vectors.
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6
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 Fencheltype duality Lov´ asz (linear) extension valuated matroid axiom, greedy algorithm integrally convex function valuated matroid intersection L/Mconvex function Fencheltype duality separation theorem M convex function L convex function polyhedral L/Mconvex function continuous L/Mconvex 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 Fencheltype minmax 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 Lconvex 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 Lconvex functions, called L convex functions, in the present theory. We have so far seen major milestones on the road toward Lconvex functions and now turn to Mconvex functions. A weighted version of the matroid intersection problem was introduced by Edmonds [44]. The problem is to ﬁnd a maximumweight 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 Fencheltype minmax duality theorem for matroid valuations (Murota [139]). This Fencheltype duality is neither a generalization nor a special case of Fujishige’s Fencheltype duality for submodular functions, but these two can be generalized into a single minmax equation, which is the Fencheltype duality theorem in the present theory. A further analogy of valuated matroids to concave functions led to the concept of Mconvex/concave functions in Murota [137], 1996. Mconvexity 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 Mconcave function deﬁned on {0, 1}vectors. The base polyhedron of an integral polymatroid is a synonym for a {0, +∞}valued Mconvex function. The valuated matroid intersection problem and the polymatroid intersection problem are uniﬁed into the Mconvex intersection problem. The Fencheltype duality theorem for matroid valuations is generalized for Mconvex functions and the submodular ﬂow problem is generalized to the Mconvex submodular ﬂow problem (Murota [142]), which involves an Mconvex function as a nonlinear cost. The nice optimality criterion using dual variables survives in this generalization. Thus, Mconvex 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 Mconvex 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. Lconvex functions form a conjugate class of Mconvex 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 Lconvex and Mconvex functions. In particular, the separation theorem for Lconvex functions is a generalization of Frank’s separation theorem for submodular functions. Ramiﬁcations of the concepts of L and Mconvexity followed. M convex functions,4 introduced by Murota–Shioura [151], are essentially equivalent to Mconvex 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 Mconvex. L convex functions, due to Fujishige–Murota [68], are an equivalent variant of Lconvex 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 Mconvex functions. This idea was worked out by Murota–Shioura [152] with the introduction of the concepts of 4 “M convex”
should be read “Mnaturalconvex” and similarly for “L convex.”
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1.2. Useful Properties of Convex Functions
9
L and Mconvexity 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 minmax 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 Lconvex or Mconvex 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 onetoone correspondence in the class of wellbehaved 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 onetoone 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 minmax 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 minmax 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 minmax 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 minmax theorem guarantees the existence of such a certiﬁcate pˆ for any optimal solution x ˆ. It is also mentioned that the minmax 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 integervalued 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 Fencheltype 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 minmax duality relation for integervalued 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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14
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 onedimensional 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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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 realvalued 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 discretemathematical 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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16
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 realvalued 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 integervalued 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 integervalued 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 integervalued 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 wellbehaved discrete convex functions, there is a fair chance of such discrete sets B being wellbehaved 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 : (BEXC[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 integervalued submodular functions ρ : 2V → Z ∪ {+∞} with ρ(∅) = 0 and ρ(V ) < +∞ and the class of nonempty subsets B ⊆ ZV satisfying (BEXC[Z]) are in onetoone 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 (BEXC[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., (BEXC[Z]) and M M0 [Z] = {B  ∅ = B ⊆ ZV , B satisﬁes (BEXC[Z])}, ˜ 0 [Z] = {δB  ∅ = B ⊆ ZV , B satisﬁes (BEXC[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 integervalued 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 onetoone 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 Mconvexity 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 minmax 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 (BEXC[Z]). Let B ⊆ ZV be a nonempty set satisfying the exchange axiom (BEXC[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 (BEXC[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 minmax 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 integervalued 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 righthand 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 integervalued 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 wellknown 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 (BEXC[Z]) is generalized to deﬁne the concept of Mconvex functions and, accordingly, Edmonds’s intersection theorem is generalized to the Fencheltype duality theorem for Mconvex 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 Lconvex and Mconvex functions are given, together with concise descriptions of their major properties, including local optimality criteria for global optimality, conjugacy between Lconvexity and Mconvexity, 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
LConvex Functions
The ﬁrst kind of discrete convex functions, Lconvex 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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Chapter 1. Introduction to the Central Concepts p∨q
q qˆ pˆ p∧q
p
Figure 1.8. Deﬁnition of Lconvexity.
(1.30) of the Lov´asz extension of a submodular set function are discretized to the following deﬁnition of Lconvex functions. We say that a function g : ZV → R ∪ {+∞} with domZ g = ∅ is Lconvex 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 Lconcave if −k is Lconvex. 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 Lconvex 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 Lconvex if it satisﬁes (SBS[Z]) and (TRS[Z]) above. Obviously, a set D is Lconvex if and only if its indicator function δD is an Lconvex function. Since an Lconvex 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 Lconvex 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 Lconvex. 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 roundup and rounddown 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 Lconvex functions, the class of L convex functions is strictly larger than that of Lconvex 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 Lconvex is the onedimensional discrete convex function depicted in Fig. 1.6 (left). 18 “L convex”
should be read “Lnaturalconvex.”
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Chapter 1. Introduction to the Central Concepts
Lconvex functions enjoy the following nice properties that are expected of discrete convex functions. • An Lconvex function can be extended to a convex function. • Local optimality (or minimality) guarantees global optimality. Speciﬁcally, we have the following: – For an Lconvex 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 Lconvex functions are endowed with the property of discreteness in direction. • Discrete duality, e.g., the Fencheltype minmax duality or discrete separation, holds good. Thus, Lconvex 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 Lconvex function and for the Fencheltype minmax duality. Lconvexity 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 integervalued distance function19 γ on V , the set of admissible integervalued potentials D = {p ∈ ZV  p(v) − p(u) ≤ γ(u, v) (∀ u, v ∈ V, u = v)}
(1.34)
is an Lconvex set. The converse is also true; i.e., any Lconvex 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 Lconvex 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 integervalued 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 Lconvex 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, Lconvexity 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 Lconvex 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 Mmatrices22 and hence they are equivalent to the (ﬁnitedimensional) Dirichlet forms known in probability theory. We conclude this section by identifying the four types of Lconvex functions that we are concerned with: realvalued Lconvex functions on integers, integervalued Lconvex functions on integers, realvalued polyhedral Lconvex functions on reals, and quadratic Lconvex functions on reals. For the ﬁrst three classes we introduce the following notation: L[Z → R] = {g : ZV → R ∪ {+∞}  g is Lconvex},
(1.36)
L[Z → Z] = {g : Z → Z ∪ {+∞}  g is Lconvex}, L[R → R] = {g : RV → R ∪ {+∞}  g is polyhedral Lconvex}.
(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 integervalued submodular set functions.
1.4.2
MConvex Functions
The second kind of discrete convex functions, Mconvex functions, is obtained from a generalization of the simultaneous exchange property (BEXC[Z]) of base polyhedra. As a motivation for the axiom of Mconvex 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. Mmatrices do not correspond to Mconvex functions but to Lconvex 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 Mconvex if it satisﬁes the following exchange axiom: (MEXC[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 righthand 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 (MEXC[Z]) can be expressed alternatively as follows: (MEXC [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 Mconcave if −h is Mconvex. It follows from (MEXC[Z]) that the eﬀective domain of an Mconvex function satisﬁes the exchange axiom (BEXC[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 Mconvexity.
for x, y ∈ domZ f in (1.40). In particular, the indicator function δB : ZV → {0, +∞} of a set B ⊆ ZV is Mconvex 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 (BEXC[Z]) as an Mconvex set . The eﬀective domain of an Mconvex function f , being an Mconvex 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 Mconvex 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 Mconvex 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 “Mnaturalconvex.” 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 Mconvex functions, the class of M convex functions is strictly larger than that of Mconvex functions. This follows from the implication (MEXC[Z]) ⇒ (M EXC[Z]). The simplest example of an M convex function that is not Mconvex is the onedimensional discrete convex function depicted in Fig. 1.6 (left). Mconvex functions enjoy the following nice properties that are expected of discrete convex functions. • An Mconvex function can be extended to a convex function. • Local optimality (or minimality) guarantees global optimality. Speciﬁcally, we have the following: – For an Mconvex 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, Mconvex functions are endowed with the property of discreteness in direction. • Discrete duality, e.g., the Fencheltype minmax duality or discrete separation, holds good. Thus, Mconvex 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 Mconvex function and for the Fencheltype minmax duality. Mconvex 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 (MEXC[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 Mconvex if it satisﬁes the following exchange property:
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29
(MEXC[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 Mconvex 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, Mconvexity 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 Mconvex functions are a quantitative generalization of the base polyhedra explained in section 1.3.2, whereas quadratic Mconvex functions are characterized in section 2.1.3 as quadratic forms deﬁned by the inverse of diagonally dominant symmetric Mmatrices. We conclude this section by identifying the four types of Mconvex functions that we are concerned with: realvalued Mconvex functions on integers, integervalued Mconvex functions on integers, realvalued polyhedral Mconvex functions on reals, and quadratic Mconvex functions on reals. For the ﬁrst three classes we introduce the following notation: M[Z → R] = {f : ZV → R ∪ {+∞}  f is Mconvex},
(1.46)
M[Z → Z] = {f : Z → Z ∪ {+∞}  f is Mconvex}, M[R → R] = {f : RV → R ∪ {+∞}  f is polyhedral Mconvex}.
(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 Lconvexity and Mconvexity is a distinguishing feature of the present theory. Whereas conjugacy in ordinary convex analysis gives a symmetric onetoone correspondence within a single class of closed proper convex functions (Theorem 1.2), conjugacy described in this section establishes a onetoone 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 integervalued Lconvex and Mconvex 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 integervalued Lconvex functions and Mconvex functions, L = L[Z → Z] and M = M[Z → Z], are in onetoone 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 Mconvex sets and their support functions, the latter being positively homogeneous Lconvex 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 Lconvex sets and by 0 M[Z → Z] that of positively homogeneous Mconvex functions. Theorem 1.14. Two classes of discrete functions, L0 = L˜0 [Z] and 0 M = 0 M[Z → Z], are in onetoone 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 Lconvex function can be identiﬁed with a submodular set function, so can a positively homogeneous Mconvex 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 onetoone 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 onetoone 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
Mconvex functions positively homogeneous Mconvex functions (Theorem 6.59) distance functions (Theorem 5.5) Lconvex sets
0M
⊂
T L0
⊂
M ⏐ ⏐ L
⊃ M0 S ⊃
0L
Lconvex functions
(base polyhedra) Mconvex sets (Theorem 4.15) submodular set functions (Theorem 7.40) positively homogeneous Lconvex 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 Mconvexity prevails in discrete systems. • In network ﬂow problems, ﬂow and tension are dual objects. Roughly speaking, ﬂow corresponds to Mconvexity and tension to Lconvexity. 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 Mconvex when expressed in terms of the terminal current supplied by current sources. It is Lconvex 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 Lconvexity and the base family to Mconvexity: rank function : L ←→ M : base family. In a valuated matroid, the valuation of bases is an Mconcave function deﬁned on the unit cube {0, 1}V . Matroids and valuated matroids are explained in section 2.4. • The concept of Mmatrices corresponds to Lconvexity. Speciﬁcally, a quadratic function is L convex if and only if it is deﬁned by a diagonally dominant symmetric Mmatrix. The inverse of such a matrix corresponds to Mconvexity. A diagonally dominant symmetric Mmatrix 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 Mconvex functions are stated here in the case of integervalued 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 Fencheltype minmax duality theorem: [Discrete Fencheltype 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 Lconvexity or Mconvexity. Then the following theorems result. Theorem 1.15 (Fencheltype duality for Lconvex 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 (Fencheltype duality for Mconvex 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 Fencheltype minmax theorem is selfconjugate. 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 Lconvexity or Mconvexity for convexity in this generic statement to obtain a conjugate pair of discrete separation theorems. Theorem 1.17 (Lseparation 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 (Mseparation 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 weightsplitting 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 Lseparation 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 Lseparation 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 Fencheltype duality theorem for Mconvex 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 Fencheltype minmax 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 weightsplitting theorem for the matroid intersection problem with integer weights is a special case of the Mseparation 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 integervalued 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 weightsplitting 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 minimumweight base of (V, B1 ) with respect to w1∗ , and (iii) B ∗ is a minimumweight 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 Mconvex and Mconcave, respectively (h is constant on B2 ). Noting that f (x) ≥ h(x) (x ∈ ZV ), as well as domZ f ∩ domZ h = ∅, we apply the Mseparation 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
Mseparation theorem f (x) ≥ α∗ + p∗ , x ≥ h(x) Fencheltype duality (Fujishige [62]) ! Intersection theorem (Edmonds [44]) ⎧ ! ⎪ ⎪ ⎪ ⇒ Discrete separation for submodular functions Fencheltype duality ⎪ ⎨ (Frank [55]) inf{f − h} ⎪ ⎪ ⇒ Valuated matroid intersection = sup{h◦ − f • } ⎪ ⎪ ⎩ (Murota [135]) ! ⇓ Lseparation 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 weightsplitting theorem, can be regarded as a special case of the Mseparation theorem (to be explained in Example 8.28). The relationship among duality theorems is summarized in Fig. 1.13. A derivation of Fujishige’s Fencheltype duality theorem from the Fencheltype duality theorem for Lconvex 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 Mconvex 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 Lconvex sets). If D1 and D2 are disjoint Lconvex 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 Mconvex sets). If B1 and B2 are disjoint Mconvex 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 convexextensible
=
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 weightsplitting theorem described in Example 1.21 is due to Frank [54]. Mconvex functions are introduced in Murota [137], followed by Lconvex functions in Murota [140]. Their fundamental properties are established in Murota [137], [140], [141], [142]; the discrete conjugacy theorem (Theorem 1.13) and the Lseparation theorem (Theorem 1.17) in [140]; the Mseparation theorem (Theorem 1.18) in [137], [140], [142]; and the Fencheltype duality theorem for Mconvex functions (Theorem 1.16) in [137], [140]. The separation theorems for Lconvex sets and Mconvex sets (Theorems 1.23 and 1.24) are due to [140]. Mconvexity
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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 + sepconv 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 + sepconv 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]
sepconv: 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 integervalued function f and Lconvexity 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 (Lconvexity) and exchangeability (Mconvexity).
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 (Lconvexity) and exchangeability (Mconvexity) 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 MMatrices
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).
[oﬀdiagonal 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 Mmatrix 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 Mmatrix is characterized as a matrix whose oﬀdiagonal entries are all nonpositive and the entries of whose inverse matrix are all nonnegative. With this terminology we can say that symmetric matrices with oﬀdiagonal nonpositivity and diagonal dominance considered in this section are exactly the same as diagonally dominant symmetric Mmatrices. In passing we mention the fact that any symmetric Mmatrix can be transformed into a diagonally dominant symmetric Mmatrix by a symmetric diagonal scaling. Mmatrices 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 Mmatrices. 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 Mmatrix. 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 oﬀdiagonal 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 positivedeﬁnite 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 oﬀdiagonal nonpositivity. Proposition 2.6. For a symmetric matrix L, the oﬀdiagonal 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 righthand side of (2.17) minus the lefthand 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 = ∅. Oﬀdiagonal nonpositivity can thus be translated into submodularity. Then how does the combination of the oﬀdiagonal 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 oﬀdiagonal 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 oﬀdiagonal 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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2.1. Quadratic Functions
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 positivedeﬁnite 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 oﬀdiagonal 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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Chapter 2. Convex Functions with Combinatorial Structures
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 matrixalgebraic 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 lefthand 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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51
(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 lefthand side is negative and the righthand 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/MConvex 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 positivesemideﬁnite 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 positivesemideﬁnite 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 positivesemideﬁnite 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 Mmatrix (with oﬀdiagonal 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
RealValued 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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Chapter 2. Convex Functions with Combinatorial Structures
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: (MEXC[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 (MEXC[R]) in the deﬁnition of Mconvex functions in section 1.4.2 and (SBF[R]) and (TRF[R]) in the deﬁnition of Lconvex 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 (Mconvexity and Lconvexity) with respect to combinatorial properties. • Exchangeability (Mconvexity) and submodularity (Lconvexity) appear as conjugate properties. The combinatorial properties (MEXC[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 integervalued ﬂows.
2.2.2
IntegerValued 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 integervalued 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: (MEXC[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 (MEXC[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 (MEXC[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 (MEXC[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 (MEXC[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 Lconvexity and Mconvexity 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 lefthand side is bounded by F (w + λχa , c) + F (w + μχb , c) and the righthand 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 linearalgebraic 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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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 lefthand 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 righthand 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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Chapter 2. Convex Functions with Combinatorial Structures
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 allimportant 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 onetoone 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 (BEXC[Z]) treated in section 1.3.2 and (R3) is the submodularity featured in section 1.3.1. The onetoone 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 lefthand side of (2.69) is equal to ω(J) + ω(J ). This implies that at least one term on the righthand 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 righthand 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 Mconvexity. 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 (MEXC[Z]) for f . That is, ω is a valuated matroid if and only if f is an Mconvex function. For instance, the valuated matroid (V, ω) associated with the polynomial matrix in (2.73) can be identiﬁed with an Mconvex 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 integervalued 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 nonArchimedian 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 − q1 ; 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 Mmatrices are wellstudied objects in applied mathematics and a comprehensive treatment of their mathematical properties can be found in Berman–Plemmons [9]. The connection of symmetric Mmatrices to L/Mconvex 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. Mconvexity and Lconvexity 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 Mconvexity and Lconvexity (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 linearalgebraic 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) halfspaces 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)
(d1 ≤ ε).
(3.25)
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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 onetoone 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 onetoone 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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83
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 (vertexoriented representation) and as the intersection of ﬁnitely many halfspaces described by linear inequalities (faceoriented 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 (BEXC[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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Chapter 3. Convex Analysis, Linear Programming, and Integrality Y = f (x)
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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85
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 minmax 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 mdimensional vector b, and an ndimensional 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 selfloops 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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3.3
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 holefree 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 holefree 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 holefree 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 holefree 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 holefree 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 holefree 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 Mconvex sets and Lconvex 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 Lconvex and Mconvex 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 convexextensible 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 integervalued 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 integervalued 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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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], HiriartUrruty–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
MConvex Sets and Submodular Set Functions
Mconvex sets form a class of wellbehaved discrete convex sets. They are deﬁned in terms of an exchange axiom and correspond onetoone to integervalued submodular set functions. An Mconvex 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 Mconvex set if it satisﬁes the following exchange axiom: (BEXC[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 Mconvex sets. Mconvexity thus deﬁned for a set B ⊆ ZV is equivalent to the Mconvexity of the indicator function δB : ZV → {0, +∞} (deﬁned in (3.51)). Namely, B is an Mconvex set satisfying (BEXC[Z]) if and only if δB is an Mconvex function satisfying (MEXC[Z]) introduced in section 1.4.2. Recall that we encountered (BEXC[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 Mconvex set is exactly the same as the set of integer points contained in the base polyhedron deﬁned by an integervalued submodular set function. An immediate consequence of the exchange axiom (BEXC[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. MConvex Sets and Submodular Set Functions
the notation x(X) =
x(v)
(x ∈ RV , X ⊆ V ),
(4.1)
v∈X
x1 =
x(v)
(x ∈ RV ).
(4.2)
v∈V
Proposition 4.1. For an Mconvex set B we have x(V ) = y(V ) for any x, y ∈ B. Proof. The proof is by induction on x − y1 . If x − y1 = 0, we obviously have x(V ) = y(V ). The case of x − y1 = 1 is excluded by (BEXC[Z]). If x − y1 ≥ 2, (BEXC[Z]) implies y ≡ y + χu − χv ∈ B, for which we have y (V ) = y(V ), x − y 1 = x − y1 − 2, and also x(V ) = y (V ) by the induction hypothesis. Since an Mconvex set lies on a hyperplane {x ∈ RV  x(V ) = r}, we may equivalently consider the projection of an Mconvex set along an arbitrarily chosen coordinate axis. We call the projection of an Mconvex set an M convex set . Whereas M convex sets are conceptually equivalent to Mconvex sets, the class of M convex sets is strictly larger than that of Mconvex sets. The simplest example of an M convex set that is not Mconvex is an integer interval [a, b]Z . We focus on Mconvex 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 (BEXC[Z]). Whereas (BEXC[Z]) requires that both x − χu + χv and y + χu − χv belong to B, (BEXC+ [Z]) below imposes this only on y + χu − χv . Proposition 4.2. For a set B ⊆ ZV , (BEXC[Z]) is equivalent to the following: (BEXC+ [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 (BEXC+ [Z]) ⇒ (BEXC[Z]). First, it is easy to see that (BEXC+ [Z]) implies the following: (BEXC−loc [Z]) For x, y ∈ B with x − y1 = 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 (BEXC[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 − y1 ; we have x − y1 ≥ 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 (BEXC+ [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 − y1 − 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 (BEXC+ [Z]). If X ∩ Y = ∅, we have / B and y + χu0 − χv ∈ / B, and therefore, y ∈ / B by (BEXC−loc [Z]). y + χu∗ − χv ∈ In either case we have (x, y ) ∈ D. We introduce two other variants: (BEXCw [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. (BEXC− [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 (BEXC[Z]), (BEXCw [Z]), (BEXC+ [Z]), and (BEXC− [Z]) are equivalent for a set B ⊆ ZV . Proof. The implication (BEXC[Z]) ⇒ (BEXCw [Z]) is obvious, and Proposition 4.2 shows (BEXC[Z]) ⇔ (BEXC+ [Z]). We also have (BEXC[Z]) ⇔ (BEXC− [Z]), since (BEXC− [Z]) for B is equivalent to (BEXC+ [Z]) for −B, and (BEXC[Z]) for B is equivalent to (BEXC[Z]) for −B. We show (BEXCw [Z]) ⇒ (BEXC− [Z]) by induction on x − y1 . Suppose x, y ∈ B and u ∈ supp+ (x − y). By (BEXCw [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 − y1 − 2 and, by the induction hypothesis, (BEXC− [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 Mconvex 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. MConvex 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 righthand 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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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 lefthand 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 X2  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 onetoone 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 onetoone 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 MConvex Sets
An Mconvex set is hole free (Theorem 4.12 below), which allows us to identify an Mconvex set with its convex hull. The convex hull of an Mconvex set is called an Mconvex polyhedron, which is indeed a polyhedron described by a submodular function (Proposition 4.13 below). Let us start with the holefree property of an Mconvex set. Theorem 4.12. B = B ∩ ZV for an Mconvex 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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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 − x1 =
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). (BEXC[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 Mconvex set is a polyhedron—the base polyhedron deﬁned by some submodular set function. Proposition 4.13. For an Mconvex 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 − z1 minimum. Then we have y(v) = z(v) (v ∈ X ∩ Y ), since, otherwise, ∃ u ∈ (X ∩ Y ) ∩ supp+ (z − y) and, by (BEXC+ [Z]), ∃ v ∈ supp− (z − y) such that y = y + χu − χv ∈ B, for which we have y (X ∪ Y ) ≥ y(X ∪ Y ) and y − z1 ≤ y − z1 − 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 Mconvex 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 (BEXC+ [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 integervalued submodular set function. (1) B = B(ρ) ∩ ZV is an Mconvex 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 (BEXC+ [Z]). Suppose, to the contrary, that (BEXC+ [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 onetoone correspondence between the family M0 [Z] of Mconvex sets and the family S[Z] of integervalued submodular set functions. In this sense, the exchange property (BEXC[Z]) and the submodularity (4.9) are equivalent. Theorem 4.15. A set B ⊆ ZV is Mconvex if and only if B = B(ρ) ∩ ZV for an integervalued 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 onetoone 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 integervalued 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 integervalued 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 X2 + y2X X2 . 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 righthand side of (4.29). On the other hand, the optimal value of Problem (A) for p = 1 is obviously equal to the lefthand 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 righthand 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 integervalued 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 righthand 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 integervalued 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
MConvex Sets as Discrete Convex Sets
We show a number of nice properties of Mconvex sets that qualify them as wellbehaved discrete convex sets. We start with a discrete separation theorem for two Mconvex sets. Theorem 4.21 (Discrete separation for Mconvex sets). Let B1 and B2 (⊆ ZV ) be Mconvex 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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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 Mconvex polyhedra. Theorem 4.22. For Mconvex 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 Mconvex sets. The claim (3) in the theorem below says that the Minkowski sum of Mconvex sets is again an Mconvex set. An important consequence of this is that the family of Mconvex 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 integervalued submodular set functions ρ1 , ρ2 ∈ S[Z], we have (B(ρ1 ) ∩ ZV ) + (B(ρ2 ) ∩ ZV ) = B(ρ1 + ρ2 ) ∩ ZV . (3) For Mconvex sets B1 , B2 ⊆ ZV , B1 + B2 is an Mconvex set and B1 + B2 = B1 + B2 ∩ ZV .
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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 lefthand side of (2) is B1 + B2 , whereas for the righthand 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 Mconvex set. Finally, we show the integral convexity of an Mconvex set. Theorem 4.24. An Mconvex set is integrally convex. Proof. Let B be an Mconvex 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 Mconvex sets is not necessarily Mconvex, 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 Mconvex 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 Mconvex sets B1 = S ∪ {(0, 1, 1, −2)} and B2 = S ∪ {(1, 1, 0, −2)}. Note that (BEXC[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 Mconvex sets are essentially equivalent, since an Mconvex set lies on a hyperplane {x ∈ RV  x(V ) = r} for some r ∈ Z (Proposition 4.1). All the results for Mconvex 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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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 Mconvex 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 Mconvex sets, the class of M convex sets is strictly larger than that of Mconvex sets. This follows from the implication (BEXC[Z]) ⇒ (B EXC[Z]), as well as from the example of an integer interval [a, b]Z that is not Mconvex 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 gpolymatroid for short); see Theorem 3.58 of Fujishige [65]. Hence, an M convex set is precisely the set of integer points of an integral gpolymatroid 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 integervalued 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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4.8
Chapter 4. MConvex Sets and Submodular Set Functions
MConvex Polyhedra
Mconvex polyhedra are deﬁned in section 4.4 as the convex hull of Mconvex sets, and as such, they are necessarily integral polyhedra. The concept of Mconvexity, however, can also be deﬁned for general (nonintegral) polyhedra. A nonempty polyhedron B ⊆ RV is deﬁned to be an Mconvex polyhedron if it satisﬁes the following: (BEXC[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: (BEXC+ [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 (BEXC[R]). That is, (BEXC[R]) ⇐⇒ (BEXC+ [R])
(4.38)
for a nonempty polyhedron B ⊆ RV (cf. Proposition 4.2). The onetoone correspondence of Mconvex sets with submodular set functions (Theorem 4.15) is generalized as follows: B is an Mconvex polyhedron ⇐⇒ B = B(ρ) for ρ ∈ S[R], (4.39) B is an integral Mconvex polyhedron ⇐⇒ B = B(ρ) for ρ ∈ S[Z], (4.40) where an integral Mconvex polyhedron means an Mconvex polyhedron B such that B = B ∩ ZV . For an integral Mconvex polyhedron B and integer points x, y ∈ B ∩ZV , we can take α0 = 1 in (BEXC[R]) by (4.40). An integral polyhedron B is Mconvex if and only if B ∩ ZV is an Mconvex set. We denote by M0 [R] the set of Mconvex polyhedra and by M0 [ZR] the set of integral Mconvex polyhedra. The projection of an Mconvex 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 [ZR] the sets of M convex polyhedra and integral M convex polyhedra, respectively.
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4.8. MConvex Polyhedra
119
An Mconvex cone means a cone that is an Mconvex 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 Mconvex 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 Mconvex cone. An Mconvex polyhedron is characterized as a polyhedron such that the tangent cone at each point is an Mconvex cone (by (a) ⇔ (b) in Theorem 6.63). Combining this with (4.42) yields a characterization of an Mconvex polyhedron in terms of the direction of edges: B is an Mconvex 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 Mconvex polyhedra and M convex polyhedra are synonyms of base polyhedra and gpolymatroids.
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 (holefree 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 Mconvex sets) is observed in Murota [140]. Integral convexity of Mconvex 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 gpolymatroids 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. MConvex 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
LConvex Sets and Distance Functions
Lconvex sets form another class of wellbehaved discrete convex sets. They are deﬁned in terms of an abstract axiom and correspond onetoone to integervalued distance functions satisfying the triangle inequality. Lconvex sets (or their convex hull) are, in fact, a familiar object in the theory of network ﬂows, though the terminology of Lconvexity is not used there. Emphasis here is placed on a systematic presentation of various properties of Lconvex sets from the viewpoint of discrete convex analysis.
5.1
Deﬁnition
A nonempty set of integer points D ⊆ ZV is deﬁned to be an Lconvex 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 Lconvex sets. Lconvexity thus deﬁned for a set D ⊆ ZV is equivalent to the Lconvexity of the indicator function δD : ZV → {0, +∞} of D, deﬁned in (3.51). Namely, D is an Lconvex set, satisfying (SBS[Z]) and (TRS[Z]), if and only if δD is an Lconvex function, satisfying (SBF[Z]) and (TRF[Z]) introduced in section 1.4.1. Since an Lconvex set is homogeneous in the direction of 1 by (TRS[Z]), we may consider the restriction of an Lconvex set to the coordinate plane deﬁned by p(v0 ) = 0 for an arbitrary v0 ∈ V . A set derived from an Lconvex set by such a restriction (intersection with a coordinate plane) is called an L convex set . Whereas L convex sets are conceptually equivalent to Lconvex sets, the class of L convex sets is strictly larger than that of Lconvex sets. The simplest example of an L convex set that is not Lconvex is an integer interval [a, b]Z . We focus on Lconvex sets in the development of the theory and deal with L convex sets in section 5.5. 121
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5.2
Chapter 5. LConvex 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 Lconvex 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 integervalued γ.
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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 righthand 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 LConvex Sets
An Lconvex set is hole free (Theorem 5.2 below), which allows us to identify an Lconvex set with its convex hull. The convex hull of an Lconvex set is called an Lconvex polyhedron, which is indeed a polyhedron described by a distance function (Proposition 5.3 below). Let us start with the holefree property of an Lconvex set. Theorem 5.2. D = D ∩ ZV for an Lconvex set D ⊆ ZV .
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Chapter 5. LConvex 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 Lconvex 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 Lconvex 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 integervalued distance function γ, D = D(γ) ∩ ZV is an Lconvex set provided that D(γ) is nonempty.
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5.4. LConvex 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 onetoone correspondence between the family L0 [Z] of Lconvex sets and the family T [Z] of integervalued distance functions with the triangle inequality. Theorem 5.5. A set D ⊆ ZV is Lconvex if and only if D = D(γ) ∩ ZV for an integervalued 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 onetoone 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 Mconvex polyhedron is described by a submodular set function (Theorem 4.15), and the correspondence is onetoone: Mconvex polyhedra ←→ submodular set functions. An Lconvex polyhedron is described by a distance function with the triangle inequality (Theorem 5.5), which gives another onetoone correspondence: Lconvex polyhedra ←→ distance functions with the triangle inequality. These two onetoone correspondences will be uniﬁed into a single conjugacy relationship between Mconvex functions and Lconvex functions in Chapter 8.
5.4
LConvex Sets as Discrete Convex Sets
We show a number of nice properties of Lconvex sets that qualify them as wellbehaved discrete convex sets. First we consider the intersection of two Lconvex 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 Lconvex 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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Chapter 5. LConvex Sets and Distance Functions ∅ ⇐⇒ no negative cycle exists in graph Gγ12 . (3) D1 ∩ D2 = (4) D1 ∩ D2 is an Lconvex 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 Lconvex sets. Theorem 5.8. For Lconvex 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 Lconvex sets. Theorem 5.9 (Discrete separation for Lconvex sets). Let D1 and D2 (⊆ ZV ) be Lconvex 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. LConvex 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 Lconvex set by deriving an expression of the convex hull of an Lconvex 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 Lconvex set can be characterized with reference to the expression (5.11). Theorem 5.10. For an Lconvex set D ⊆ ZV , we have D = {p ∈ RV  p + χUi (p) ∈ D (i = 0, 1, . . . , m)}.
(5.12)
Hence, an Lconvex 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 integervalued 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. LConvex 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 Lconvex sets, to be called an L2 convex set , is not necessarily Lconvex, though it is integrally convex (see Theorem 8.42). An example of an L2 convex set that is not Lconvex 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 Lconvex 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 Lconvex sets are essentially equivalent, since an Lconvex set is homogeneous in the direction of 1 by (TRS[Z]). All the results for Lconvex 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 Lconvex 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 Lconvex sets, the class of L convex sets is strictly larger than that of Lconvex 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 Lconvex 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 roundup and rounddown 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 integervalued distance function γ : V × V → Z ∪ {+∞} and integervalued 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. LConvex 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. LConvex Polyhedra
5.6
131
LConvex Polyhedra
Lconvex polyhedra are deﬁned in section 5.3 as the convex hull of Lconvex sets, and as such they are necessarily integral polyhedra. The concept of Lconvexity, however, can also be deﬁned for general (nonintegral) polyhedra. A nonempty polyhedron D ⊆ RV is deﬁned to be an Lconvex 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 Lconvex polyhedron we mean an Lconvex polyhedron D such that D = D ∩ ZV . An integral polyhedron D is Lconvex if and only if D ∩ ZV is an Lconvex set. We denote by L0 [R] the set of Lconvex polyhedra and by L0 [ZR] the set of integral Lconvex 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 Lconvex polyhedron. The onetoone correspondence of Lconvex sets with distance functions (Theorem 5.5) is generalized as follows: ⇐⇒ D = D(γ) for γ ∈ T [R], (5.18)
D is an Lconvex polyhedron
D is an integral Lconvex polyhedron ⇐⇒ D = D(γ) for γ ∈ T [Z]. (5.19) The restriction of an Lconvex 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 [ZR] the sets of L convex polyhedra and integral L convex polyhedra, respectively. An Lconvex cone means a cone that is an Lconvex polyhedron. We have ) ( * ) ) D is an Lconvex 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 Lconvex polyhedron is characterized as a polyhedron such that the tangent cone at each point is an Lconvex cone (by (a) ⇔ (b) in Theorem 7.45).
Bibliographical Notes The concept of Lconvex 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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132
Chapter 5. LConvex Sets and Distance Functions
function is a wellstudied object, appearing, e.g., in the dual of the transshipment problem. In particular, (5.5) is known as the maximumseparation minimumroute 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 Lconvex sets by D(γ)) and Theorem 5.9 (separation of Lconvex sets) are due to Murota [140]. Theorem 5.2 (holefree property), Theorem 5.7 (convexity in intersection), and Theorem 5.8 (convexity in Minkowski sum) are by Murota [141]. Integral convexity of Lconvex 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 Lconvex polyhedra in section 5.6 are considered in Murota–Shioura [152].
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Chapter 6
MConvex Functions
Mconvex functions form a class of wellbehaved discrete convex functions. They are deﬁned in terms of an exchange axiom and are characterized as functions obtained by piecing together Mconvex sets in a consistent way or as collections of distance functions with some consistency. Fundamental properties of Mconvex 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
MConvex Functions and M Convex Functions
We recall the deﬁnitions of Mconvex functions and M convex functions from section 1.4.2. A function f : ZV → R ∪ {+∞} with dom f = ∅ is said to be an Mconvex function if it satisﬁes the following exchange axiom: (MEXC[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 (MEXC[Z]) can be expressed alternatively as follows: (MEXC [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. MConvex Functions
where the maximum and the minimum over an empty set are −∞ and +∞, respectively. We denote by M[Z → R] the set of Mconvex functions and by M[Z → Z] the set of integervalued Mconvex functions. Proposition 6.1. The eﬀective domain of an Mconvex function is an Mconvex set. Therefore, it lies on a hyperplane {x ∈ RV  x(V ) = r} for some integer r. Proof. It follows from (MEXC[Z]) that B = dom f satisﬁes (BEXC[Z]). Then the latter half follows from Proposition 4.1. Since the eﬀective domain of an Mconvex 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 Mconvex 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 Mconvex function. We denote by M [Z → R] the set of M convex functions and by M [Z → Z] the set of integervalued 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. (MEXC[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 Mconvex functions, but the class of M convex functions is larger than that of Mconvex functions. Theorem 6.3. An Mconvex function is M convex. Conversely, an M convex function is Mconvex 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 (MEXC[Z])⇒(M EXC[Z]) and Theorem 6.2. The second half is from the equivalence of (MEXC[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 Mconvex 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 Mconvex functions and M convex functions, all theorems stated for Mconvex functions can be rephrased for M convex functions, and vice versa. In this book we primarily work with Mconvex functions, making explicit statements for M convex functions when appropriate.
6.2
Local Exchange Axiom
There are a number of axioms equivalent to (MEXC[Z]). We consider here a local exchange axiom: (MEXCloc [Z]) For x, y ∈ dom f with x − y1 = 4, there exist u ∈ supp+ (x − y) and v ∈ supp− (x − y) such that (6.1) holds true.
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136
Chapter 6. MConvex Functions
On expressing y = x − χu1 − χu2 + χv1 + χv2 with u1 , u2 , v1 , v2 ∈ V and {u1 , u2 } ∩ {v1 , v2 } = ∅, we see that (MEXCloc [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 Mconvex set, then (MEXC[Z]) ⇐⇒ (MEXCloc [Z]). Proof. It suﬃces to show that (MEXCloc [Z]) ⇒ (MEXC[Z]). To prove this by contradiction, we assume that there exists a pair (x, y) for which (MEXC[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 − y1 , where x − y1 > 4 by (MEXCloc [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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6.2. Local Exchange Axiom
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 − y1 > 4. By (BEXC[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 (MEXCloc [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 (MEXC[Z]) is equivalent to a weak exchange axiom: (MEXCw [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 (MEXC[Z]) and “∃ u, ∃ v” in (MEXCw [Z]). Theorem 6.5. (MEXC[Z]) ⇐⇒ (MEXCw [Z]). Proof. It suﬃces to show ⇐ when dom f = ∅. (MEXCw [Z]) for f implies (BEXCw [Z]) for dom f , and therefore, dom f is an Mconvex 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 (MEXC[Z]) for f˜ deﬁned by (6.4). First, dom f˜ is an Mconvex 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, (MEXC[Z]) is equivalent to (MEXCloc [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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138
Chapter 6. MConvex Functions
(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 Mconvexity in network ﬂows and in matroids (section 2.2, section 2.4). In this section we see some other examples of Mconvex 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 Mconvex function is an Mconvex 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 Mconvex or M convex according as dom f is Mconvex or M convex. The inequalities (6.1) in (MEXC[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 Mconvex if dom f is an Mconvex 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 positivedeﬁnite 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 Mconvex 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 righthand 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). Mconvex functions admit the following operations. Theorem 6.13. Let f, f1 , f2 ∈ M[Z → R] be Mconvex functions. (1) For λ ∈ R++ , λf is Mconvex. (2) For a ∈ ZV , f (a − x) and f (a + x) are Mconvex in x. (3) For p ∈ RV , f [−p] is Mconvex. (4) For ϕv ∈ C[Z → R] (v ∈ V ), f˜(x) = f (x) + ϕv (x(v)) (x ∈ ZV )
(6.46)
v∈V
is Mconvex provided dom f˜ = ∅. (5) For a, b ∈ (Z ∪ {±∞})V , the restriction f[a,b] to the integer interval [a, b] is Mconvex provided dom f[a,b] = ∅. (6) For U ⊆ V , the restriction fU is Mconvex provided dom fU = ∅. (7) For U ⊆ V , the aggregation f U∗ is Mconvex provided f U∗ > −∞. (8) The integer inﬁmal convolution f˜ = f1 2Z f2 is Mconvex provided f˜ > −∞. Proof. (1), (2), (5), and (6) are obvious and (3) is a special case of (4).
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Chapter 6. MConvex Functions
(4) For x, y ∈ dom f˜ ⊆ dom f and u ∈ supp+ (x − y), use (MEXC[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 Mconvex ⇐⇒ f[a,b] is Mconvex 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 Mconvex functions is not necessarily Mconvex. For example, recall the Mconvex sets B1 and B2 in Note 4.25 such that B1 ∩ B2 is not Mconvex. Their indicator functions are Mconvex, but their sum, which is the indicator function of B1 ∩ B2 , is not Mconvex. The sum of two Mconvex 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]). Mconvexity (or M convexity) is not preserved under scaling. For example, the indicator function f of an Mconvex 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 Mconvex, 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 Mconvex. 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 Mconvex functions). It is worth mentioning that some subclasses of Mconvex functions are closed under the scaling operation; linear, quadratic, separable, and laminar Mconvex functions form such subclasses. See Proposition 10.41 for a type of scaling operation for Mconvex 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 − y1 , where we may assume supp+ (x − y) = ∅ and supp− (x − y) = ∅. By (6.49), (6.48) is true if x − y1 ≤ 2. For x, y with
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Chapter 6. MConvex Functions
x − y1 ≥ 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 − y1 ≤ x − y1 − 1 and x − y 1 = x − y1 − 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 Mconvex function f because x ∨ y, x ∧ y ∈ dom f occurs only when x = y. For an Mconvex 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 Mconvex 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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Proposition 6.23. An Mconvex function f ∈ M[Z → R] satisﬁes (6.50). Proof. By (MEXC[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 (MEXC[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−y1 /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 Mconvexity. For an Mconvex function f and any p ∈ RV , f [p] is again Mconvex, and, therefore, f satisﬁes the following property: (MSI[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 Mconvex function ⇐⇒ f satisﬁes (MSI[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 Mconvex 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 (MSI[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 (BEXC− [Z]) for B. Then B is an Mconvex set by Theorem 4.3. Claim 2: f satisﬁes the local exchange axiom (MEXCloc [Z]). (Proof of Claim 2) Take x, y ∈ B with x − y1 = 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 (BEXC[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 righthand 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 (MSI[Z]). The proof of Proposition 6.23 shows the following. Proposition 6.25. For an Mconvex 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 Mconvex function is characterized by local optimality. Theorem 6.26 (Moptimality criterion). (1) For an Mconvex 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 Moptimality 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 wellknown optimality criterion is a special case of Theorem 6.26 (1) applied to an Mconvex 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 Mconvex function minimization, to be explained in section 10.1.3. Theorem 6.28 (Mminimizer cut). Let f : ZV → R ∪ {+∞} be an Mconvex 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 (MEXC[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, (MEXC[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 Mconvex function form an Mconvex set, a property that is essential for a function to be Mconvex. Proposition 6.29. For an Mconvex function f ∈ M[Z → R], arg min f is an Mconvex 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 (BEXC[Z]). The following theorem reveals that Mconvex functions are characterized as functions obtained by piecing together Mconvex sets in a consistent way. This
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shows how the concept of Mconvex 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 Mconvex ⇐⇒ arg min f [−p] is an Mconvex 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 Mconvex set and f satisﬁes the local exchange axiom (MEXCloc [Z]). Claim 1: B is an Mconvex 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 (BEXC+ [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 (BEXC+ [R]) for B. Therefore, B is an Mconvex set. For (MEXCloc [Z]), take x, y ∈ B with x−y1 = 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 Mconvex polyhedron. Hence, the intersection of Bp with the interval I = [x ∧ y, x ∨ y]R is an integral Mconvex 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−y1 = 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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Chapter 6. MConvex Functions
Finally we derive (MEXCloc [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 (BEXC[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 Mconvex if and only if its restriction f[a,b] to every bounded integer interval [a, b] is Mconvex (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 Mconvex, but, for each p ∈ RV , arg min f [−p] is equal to {0} (an Mconvex set) if it is not empty.
6.8
Gross Substitutes Property
In the previous section we saw that the minimizers of an Mconvex function f form an Mconvex 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 Mconvexity, 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 wellknown 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: (MGS[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 Mconvex function f ∈ M[Z → R] satisﬁes (MGS[Z]). Proof. For x ∈ arg min f [p] and p ≤ q, let y be an element of arg min f [q] with y − x1 minimum. Suppose, on the contrary, that p(u) = q(u) and x(u) > y(u) for some u ∈ V . By (MEXC[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 (MGS[Z]) if and only if f satisﬁes the following:
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154
Chapter 6. MConvex 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]) ⇒ (MGS[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 (MGS[Z]) and (M GS[Z]) characterize Mconvex 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 Mconvex ⇐⇒ f satisﬁes (MGS[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 Mconvex set, since (MGS[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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6.8. Gross Substitutes Property
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 Moptimality 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 convexextensible 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 Mconvex 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 Mconvex function is not necessarily Mconvex, and hence a local optimum of f α may not be a global optimum of f α . The following proximity theorem, named the Mproximity 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 (Mproximity theorem). Assume α ∈ Z++ and n = V . (1) Let f : ZV → R ∪ {+∞} be an Mconvex 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 (MEXC[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}. (MEXC[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 Mproximity 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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Chapter 6. MConvex Functions
Theorem 6.39 (Mminimizer cut with scaling). Let f : ZV → R ∪ {+∞} be an Mconvex 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, Mproximity 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 Mproximity 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 Mconvex 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 Mconvex functions; note that the exchange axiom (MEXC[Z]) refers only to function values on integer points. The convex extension of an Mconvex function can be obtained by piecing together Mconvex polyhedra in a consistent way.
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159
The ﬁrst theorem shows that the convex extension of an Mconvex 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 Mconvex function f . The restriction of f to any bounded integer interval [a, b], denoted by f[a,b] , is an Mconvex function (Proposition 6.14). For any p ∈ RV , arg min(f[a,b] [−p]) is an Mconvex 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 Mconvex function as a collection of Mconvex 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 Mconvex ⇐⇒ (ii) for every p ∈ RV , arg min f [−p] is ⎩ an Mconvex 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 Mconvex 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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Chapter 6. MConvex Functions
Proof. We may assume f1 and f2 to be Mconvex 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 Mconvex 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 MConvex Functions
As we have seen, Mconvex functions on the integer lattice can be extended to convex functions in real variables. The convex extension of an Mconvex function is a polyhedral convex function when restricted to a ﬁnite interval. Motivated by this we deﬁne here the concept of Mconvexity for polyhedral convex functions in general and show that major properties of Mconvex functions survive in this generalization. A polyhedral convex function f : RV → R ∪ {+∞} with domR f = ∅ is said to be Mconvex if it satisﬁes the following exchange property: (MEXC[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)), (MEXC[R]) can be rewritten as follows: (MEXC [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 Mconvex functions. Polyhedral Mconcave functions are deﬁned in an obvious way. An Mconvex 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 Mconvex function f ∈ M[Z → R] on the integer lattice is a polyhedral Mconvex 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 Mconvex 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 dualintegral 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[ZR → R] and C[R → RZ] the sets of univariate polyhedral convex functions with integrality (6.75) and dual integrality (6.76), respectively. Polyhedral Mconvex functions with integrality (6.75) are referred to as integral polyhedral Mconvex functions, the set of which is denoted by M[ZR → R]. Polyhedral Mconvex functions with dual integrality (6.76) are referred to as dualintegral polyhedral Mconvex functions, the set of which is denoted by M[R → RZ]. By Theorems 6.45 and 6.43, an integral polyhedral Mconvex function is nothing but a polyhedral Mconvex function that can be obtained as the convex extension of an Mconvex function on integer points. Therefore, we have M[ZR → R] ⊆ M[R → R],
M[ZR → R] → M[Z → R],
(6.77)
where the second expression means that there exists an injection from M[ZR → R] to M[Z → R], representing an embedding of M[ZR → R] into M[Z → R]. The eﬀective domain of a polyhedral Mconvex function is an Mconvex 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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Chapter 6. MConvex Functions
Mconvex functions via (6.4). We denote by M [R → R] the set of polyhedral M convex functions and by M [ZR → 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 Mconvex functions and polyhedral M convex functions in n variables. The following are the Rcounterparts 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 Mconvex function is polyhedral M convex. Conversely, a polyhedral M convex function is polyhedral Mconvex if and only if the eﬀective domain is contained in {x ∈ RV  x(V ) = r} for some r ∈ R. Almost all properties of Mconvex functions on integer points carry over to polyhedral Mconvex 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 Mconvex functions. (1) For λ ∈ R++ , λf is polyhedral Mconvex. (2) For a ∈ RV and β ∈ R \ {0}, f (a + βx) is polyhedral Mconvex in x. (3) For p ∈ RV , f [−p] is polyhedral Mconvex.
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6.11. Polyhedral MConvex 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 Mconvex provided domR f˜ = ∅. (5) For a, b ∈ (R ∪ {±∞})V , the restriction f[a,b] to the real interval [a, b] is polyhedral Mconvex provided domR f[a,b] = ∅. (6) For U ⊆ V , the restriction fU is polyhedral Mconvex provided domR fU = ∅. (7) For U ⊆ V , the aggregation f U∗ is polyhedral Mconvex provided U∗ f > −∞. (8) The inﬁmal convolution f˜ = f1 2 f2 is polyhedral Mconvex 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 (Moptimality criterion). (1) For a polyhedral Mconvex 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 Mconvex function. For any p ∈ RV , arg min f [−p] is an Mconvex polyhedron if it is not empty. The property in Proposition 6.53 characterizes polyhedral Mconvexity, to be shown in Theorem 6.63. Note 6.54. Here are two remarks on α0 in (MEXC[R]). First, for an integral polyhedral Mconvex function f ∈ M[ZR → R], we can take α0 = 1. Second, if (MEXC[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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Chapter 6. MConvex Functions
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 MConvex Functions
There exists a onetoone correspondence between positively homogeneous Mconvex functions and distance functions satisfying the triangle inequality. We denote by 0 M[R → R] the set of polyhedral Mconvex functions that are positively homogeneous in the sense of (3.32) and by 0 M[ZR → R] the set of integral polyhedral Mconvex functions that are positively homogeneous. Also we denote by 0 M[Z → R] the set of Mconvex 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[ZR → R] = 0 M[R → R],
(6.80)
by the following proposition. We introduce yet another notation, 0 M[Z → Z], for the set of integervalued functions belonging to 0 M[Z → R]. Proposition 6.56. (1) 0 M[ZR → 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 Mconvex 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[ZR → 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 Mconvex 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 realvalued and integervalued 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 (MEXC[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 (MEXC[Z]) in a similar manner.
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165
Conversely, a distance function satisfying the triangle inequality induces a positively homogeneous Mconvex 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 Lconvex 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 Mconvex 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 onetoone correspondence between positively homogeneous Mconvex 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 onetoone 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 Mconvex functions are considered in this section. For a polyhedral Mconvex function f , the directional derivative f (x; d) is a positively homogeneous Mconvex function in d, and the subgradients of f at a point form an Lconvex polyhedron. Furthermore, each of these properties characterizes Mconvexity. We start with directional derivatives of a polyhedral Mconvex function f ∈ M[R → R]. Recall from (3.25) that, for each x ∈ domR f , there exists ε > 0 such that (d1 ≤ 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 integervalued 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 Mconvex functions are given as follows. It is recalled that L0 [R], L0 [ZR], L0 [Z], and M[R → RZ] denote, respectively, the sets of Lconvex polyhedra, integral Lconvex polyhedra, Lconvex sets, and dualintegral polyhedral Mconvex 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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and ∂R f (x) = ∅ in particular. If f ∈ M[R → RZ], then γf,x ∈ T [Z],
∂R f (x) ∈ L0 [ZR].
(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 [ZR],
∂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 (Moptimality 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 → RZ], ∂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 (MEXC[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[ZR → 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 Mconvex functions in terms of the Mconvexity of directional derivatives, the Lconvexity of subdiﬀerentials, and the Mconvexity 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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Chapter 6. MConvex Functions Y 6
Y = f (x) x Figure 6.3. Quasiconvex 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 Mconvex 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[ZR → R]. (d) arg min f [−p] ∈ M0 [ZR] 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 MConvex Functions
Quasi Mconvex functions are introduced as a generalization of Mconvex 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) quasiconvex 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. Quasiconvex functions enjoy the following nice properties: • A strict local minimum of a quasiconvex function is a strict global minimum. • A local minimum of a semistrictly quasiconvex function is a global minimum. • Level sets of quasiconvex 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 Mconvexity is deﬁned for a function f : ZV → R∪{+∞} as follows. Recall the exchange axiom for Mconvex functions: (MEXC[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 Mconvex functions. We say that a function f : ZV → R ∪ {+∞} with dom f = ∅ is quasi Mconvex 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 Mconvex 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 Mconvex function arises from a nonlinear scaling of an Mconvex function. For an Mconvex 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 − y1 . Suppose x, y ∈ dom f and f (x) ≥ f (y); we may assume x−y1 > 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 Mconvex 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) (MEXC[Z]) =⇒ (SSQM) =⇒ (QM) ! ⇓ ⇓ (MEXCw [Z]) =⇒ (SSQMw ) =⇒ (QMw ) (2) f satisﬁes (MEXC[Z]) ⇐⇒ ∀ p ∈ RV : f [p] satisﬁes (QMw ). Proof. (1) The equivalence of (MEXC[Z]) and (MEXCw [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 Mconvexity 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: (QEXC) 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. (QEXCw ) 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 (QEXCw ) 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 (QEXCw ), 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 (BEXC[Z]) ⇒ (QEXC) ⇒ (QEXCw ), these conditions are not equivalent. For instance, B1 = {χS  S = {1, 2}, {2, 3}, {3, 4}} satisﬁes (QEXC) and not (BEXC[Z]), and B2 = {χS  S = {1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {4, 5, 6}} satisﬁes (QEXCw ) and not (QEXC).
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The weaker version (QMw ) of quasi Mconvexity for functions can be characterized by the corresponding quasi Mconvexity (QEXCw ) 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 (QEXCw ) 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 (QEXCw ) and Proposition 6.69. Hence f (x − χu + χv ) ≤ f (x).
Proposition 6.73. If f satisﬁes (QMw ), then dom f satisﬁes (QEXCw ). Proof. In the proof of “only if” of Theorem 6.72, replace L(f, α) with dom f . An Mconvex function can be characterized by quasi Mconvexity of level sets of perturbed functions. Theorem 6.74. A function f : ZV → R ∪ {+∞} satisﬁes (MEXC[Z]) if and only if the level set L(f [p], α) satisﬁes (QEXCw ) 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 (MEXCloc [Z]) are equivalent if dom f satisﬁes (QEXCw ). (This can be shown by modifying the proof of Claim 2 in the proof of Theorem 6.4.) Note that (QEXCw ) holds for dom f by Theorem 6.72 and Proposition 6.73. To show (MEXCloc [Z]), take x, y ∈ dom f with x−y1 = 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 (QEXCw ) 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 (QEXCw ) 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 (QEXCw ) for L(f [p], f [p](x)) yields (6.11). The remaining cases are similar. Next we turn to the minimization of quasi Mconvex 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 Mconvex function is characterized by local optimality. Theorem 6.76 (Quasi Moptimality 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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Chapter 6. MConvex Functions
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 Mconvex functions can be generalized for quasi Mconvex functions. Theorem 6.77 (Quasi Mminimizer 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 (MEXC[Z]) is replaced with (SSQM = ). The proximity theorem for Mconvex functions (Theorem 6.37) can be generalized for quasi Mconvex functions. Theorem 6.78 (Quasi Mproximity 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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175
(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 Mminimizer 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 Mconvex 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 Mconvex 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 Mconvexity is noted by A. Shioura. Quasiseparable 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 minimumvalue functions (6.36) is due to Danilov–Koshevoy–Murota [34], [35]. The names laminar convex functions and minimumvalue 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 Mconvex functions is considered by Moriguchi–Murota–Shioura [133].
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176
Chapter 6. MConvex Functions
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 (Mminimizer 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 (Mproximity theorem) is by Moriguchi–Murota–Shioura [133]. Theorem 6.39 (Mminimizer cut with scaling) is by Tamura [197]. The convex extension of Mconvex 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 Mconvex functions (Theorem 6.44) are by Murota–Shioura [153]. Polyhedral Mconvex 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. Mconvexity for nonpolyhedral convex functions is considered in Murota–Shioura [156], [157]. The correspondence between positively homogeneous Mconvex 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 Mconvex 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 Mconvex functions by level sets) by Shioura [191] and Theorem 6.79 (quasi Mminimizer cut with scaling) by Tamura [197]. Zimmermann [221] considers combinatorial optimization problems with quasiconvex objective functions in real variables. Mconvex 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
LConvex Functions
Lconvex functions form another class of wellbehaved discrete convex functions. They are deﬁned in terms of an abstract axiom involving submodularity and are characterized as functions obtained by piecing together Lconvex sets in a consistent way or as collections of submodular set functions with some consistency. Fundamental properties of Lconvex 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
LConvex Functions and L Convex Functions
We recall the deﬁnitions of Lconvex functions and L convex functions from section 1.4.1. A function g : ZV → R ∪ {+∞} with dom g = ∅ is said to be an Lconvex 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 Lconvex functions and by L[Z → Z] the set of integervalued Lconvex functions. Since an Lconvex 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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178
Chapter 7. LConvex 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 Lconvex 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 Lconvex 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 integervalued 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 Lconvex functions, but the class of L convex functions is larger than that of Lconvex 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 Lconvex 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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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 Lconvex function is L convex. Conversely, an L convex function is Lconvex 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 Lconvex 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 Lconvex functions and L convex functions all theorems stated for Lconvex functions can be rephrased for L convex functions, and vice versa. In this book we primarily work with Lconvex 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. LConvex 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 Lconvex 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 Lconvexity in network ﬂows and in matroids (section 2.2, section 2.4). In this section we see some other examples of Lconvex 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 Lconvex function is an Lconvex 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. LConvex Functions A linear (or aﬃne) function48 g(p) = α + p, x
(p ∈ dom g)
(7.8)
with x ∈ Rn and α ∈ R is Lconvex or L convex according as dom g is Lconvex 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 Lconvex 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 realvalued 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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7.4. Basic Operations
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 Lconvex if dom g is an Lconvex 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 Lconvex. Maximumcomponent 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 Lconvex. 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 Lconvex functions are presented here, whereas the most important operation, transformation by networks, is treated later in section 9.6. Lconvex 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 Lconvex functions. (1) For λ ∈ R++ , λg is Lconvex. (2) For a ∈ ZV and β ∈ Z \ {0}, g(a + βp) is Lconvex in p. (3) For x ∈ RV , g[−x] is Lconvex. (4) For U ⊆ V , the projection g U is Lconvex 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. LConvex Functions
is Lconvex provided g˜ > −∞. (6) The sum g1 + g2 is Lconvex 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 Lconvexity 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 Mconvex 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 Lconvex functions is not necessarily Lconvex, 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 Lconvex function is characterized by local optimality. Theorem 7.14 (Loptimality criterion). (1) For an Lconvex 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 Lconvex function is L convex (Theorem 7.3) and the righthand side of (7.19) implies g(p − χY ) = g(p + χV \Y ) ≥ g(p). The wellknown 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 Lconvex function form an Lconvex set, a property that is essential for a function to be Lconvex.
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Chapter 7. LConvex Functions
Proposition 7.16. For an Lconvex function g ∈ L[Z → R], arg min g is an Lconvex 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 Lconvex functions are characterized as functions obtained by piecing together Lconvex sets in a consistent way. This shows how the concept of Lconvex functions can be deﬁned from that of Lconvex sets. Theorem 7.17. Let g : ZV → R ∪ {+∞} be a function with a bounded nonempty eﬀective domain. (1) g is Lconvex ⇐⇒ arg min g[−x] is an Lconvex 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 Lconvex function minimization, stating that a global optimum of an Lconvex 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 Lconvex by Theorem 7.10 (2), and accordingly, a local minimizer of g α is a global minimizer of g α by the Loptimality criterion (Theorem 7.14). Theorem 7.18 (Lproximity theorem). Assume α ∈ Z++ and n = V . (1) Let g : ZV → R ∪ {+∞} be an Lconvex 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 Lconvex 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 Lconvex functions in terms of axioms referring only to function values on integer points. The convex extension of an Lconvex 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 Lconvex, 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. LConvex Functions
The next theorem shows that an Lconvex 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 Lconvex 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 righthand 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 righthand 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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189
(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 Lconvex 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 LConvex Functions
As we have seen, Lconvex functions on the integer lattice can be extended to convex functions in real variables. The convex extension of an Lconvex function is a polyhedral convex function when restricted to a ﬁnite interval. Motivated by this
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Chapter 7. LConvex Functions
we deﬁne here the concept of Lconvexity for polyhedral convex functions in general and show that major properties of Lconvex functions survive in this generalization. A polyhedral convex function g : RV → R ∪ {+∞} with domR g = ∅ is said to be Lconvex 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 realvector lattice and (TRF[R]) is linearity in the direction of 1. We denote by L[R → R] the set of polyhedral Lconvex functions. Polyhedral Lconcave 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 Lconvex 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 ﬁnitevalued case. For each k ∈ Z++ , we deﬁne ρk by ρk (X) = min {ρ(X \ Y ) + kY } 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 Lconvex 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 Lconvex function g ∈ L[Z → R] on the integer lattice is a polyhedral Lconvex 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 Lconvex 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 Lconvex functions with integrality (6.75) are referred to as integral polyhedral Lconvex functions, the set of which is denoted by L[ZR → R]. Polyhedral Lconvex functions with dual integrality (6.76) are referred to as dualintegral polyhedral Lconvex functions, the set of which is denoted by L[R → RZ].
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Chapter 7. LConvex Functions
By Theorem 7.26 and Proposition 7.16 an integral polyhedral Lconvex function is nothing but a polyhedral Lconvex function that can be obtained as the convex extension of an Lconvex function on integer points. Therefore, we have L[ZR → R] ⊆ L[R → R],
L[ZR → R] → L[Z → R],
(7.31)
where the second expression means that there exists an injection from L[ZR → R] to L[Z → R], representing an embedding of L[ZR → R] into L[Z → R]. The eﬀective domain of a polyhedral Lconvex function is an Lconvex polyhedron, which is homogeneous in direction 1. Hence, polyhedral L convex functions can be deﬁned as the restriction of polyhedral Lconvex functions, just as L convex functions on integer points are deﬁned from Lconvex functions via (7.2). We denote by L [R → R] the set of polyhedral L convex functions and by L [ZR → 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 Lconvex functions and polyhedral L convex functions in n variables. The Rcounterpart 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 Lconvex 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 Lconvex function is polyhedral L convex. Conversely, a polyhedral L convex function is polyhedral Lconvex if and only if it satisﬁes (TRF[R]). Almost all properties of Lconvex functions on integer points carry over to polyhedral Lconvex 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 Lconvex functions. (1) For λ ∈ R++ , λg is polyhedral Lconvex. (2) For a ∈ RV and β ∈ R \ {0}, g(a + βp) is polyhedral Lconvex in p. (3) For x ∈ RV , g[−x] is polyhedral Lconvex. (4) For U ⊆ V , the projection g U is polyhedral Lconvex provided g U > −∞.
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7.9. Positively Homogeneous LConvex 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 Lconvex provided g˜ > −∞. (6) The sum g1 + g2 is polyhedral Lconvex 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 (Loptimality criterion). (1) For a polyhedral Lconvex 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 Lconvex function. For any x ∈ RV , arg min g[−x] is an Lconvex polyhedron if it is not empty. The property in Proposition 7.34 characterizes polyhedral Lconvexity, 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 LConvex Functions
Positively homogeneous Lconvex functions coincide with the Lov´ asz extensions of submodular set functions. We denote by 0 L[R → R] the set of polyhedral Lconvex functions that are positively homogeneous in the sense of (3.32) and by 0 L[ZR → R] the set of
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integral polyhedral Lconvex functions that are positively homogeneous. Also we denote by 0 L[Z → R] the set of Lconvex 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[ZR → R] = 0 L[R → R],
(7.34)
by the following proposition. We introduce yet another notation, 0 L[Z → Z], for the set of integervalued functions belonging to 0 L[Z → R]. Proposition 7.37. (1) 0 L[ZR → 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 Lconvex 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[ZR → 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 Lconvex 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 realvalued and integervalued 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 Lconvex 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 integervalued ρ.
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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 onetoone correspondence between positively homogeneous Lconvex 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 onetoone 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 Lconvex 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 Lconvex functions are considered in this section. For a polyhedral Lconvex function g, the directional derivative g (p; d) is a positively homogeneous Lconvex function in d and the subgradients of g at a point form an Mconvex polyhedron. Furthermore, each of these properties characterizes Lconvexity. We start with directional derivatives of a polyhedral Lconvex 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 Lconvex functions are given as follows. It is recalled that M0 [R], M0 [ZR], M0 [Z], and L[R → RZ] denote, respectively, the sets of Mconvex polyhedra, integral Mconvex polyhedra, Mconvex sets, and dualintegral polyhedral Lconvex 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 → RZ], then ρg,p ∈ S[Z],
∂R g(p) ∈ M0 [ZR].
(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 [ZR],
∂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) (Loptimality 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 → RZ], ∂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[ZR → 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 Lconvex functions in terms of the Lconvexity of directional derivatives, the Mconvexity of subdiﬀerentials, and the Lconvexity 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 onetoone 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 Lconvex 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 Lconvex 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[ZR → R]. (d) arg min g[−x] ∈ L0 [ZR] 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 Lconvex 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 [ZR]. This implies g ∈ L[ZR → R] by Theorem 7.46 and therefore g ∈ L[Z → R].
7.11
Quasi LConvex Functions
Quasi Lconvex functions are introduced as a generalization of Lconvex functions. The optimality criterion and the proximity theorem survive in this generalization. To deﬁne quasi Lconvexity, 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 Lconvex if it satisﬁes (QSB) and (TRF[Z]) and semistrictly quasi Lconvex if it satisﬁes (SSQSB) and (TRF[Z]). Example 7.48. A quasi Lconvex function arises from a nonlinear scaling of an Lconvex 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 Lconvexity, most of the properties of quasisubmodular functions can be restated naturally in terms of quasi Lconvex functions, and vice versa. We will work mainly with quasisubmodular functions. The following are quasi versions of Theorem 7.2 for Lconvex functions.
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Chapter 7. LConvex Functions
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 quasisubmodular 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 Lconvex function. We assume r = 0 in (TRF[Z]) since otherwise no minimizer exists. Global minimality is characterized by local minimality. Theorem 7.53 (Quasi Loptimality 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 Lconvex functions (Theorem 7.18) can be generalized for quasi Lconvex functions. Theorem 7.54 (Quasi Lproximity 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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Chapter 7. LConvex Functions
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 Lconvex functions was introduced by Murota [140]. L convex functions are deﬁned by Fujishige–Murota [68] as a variant of Lconvex 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 Lconvex 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 Lconvex functions are of fundamental importance. Theorem 7.14 (Loptimality 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 Lproximity 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 Lconvex 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 Lconvex 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. Lconvexity for nonpolyhedral convex functions is considered in Murota–Shioura [156], [157]. The correspondence between positively homogeneous Lconvex 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 Lconvex 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 onetoone correspondence within a single class of closed convex functions, conjugacy in discrete convex analysis establishes a onetoone correspondence between two diﬀerent classes of discrete functions with diﬀerent combinatorial properties distinguished by “L” and “M.” The conjugacy between Lconvexity and Mconvexity 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 Mconvex/Mconcave functions and for Lconvex/Lconcave functions (M and Lseparation theorems) and the Fencheltype 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
Mconvex functions and Lconvex 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 onetoone correspondence within a single class of closed convex functions. The conjugacy correspondence between Mconvexity and Lconvexity 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/Lconvex functions is established in section 8.1.2 and that for integervalued M/Lconvex functions on integer points in section 8.1.3. 205
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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 lefthand side of (8.5) over x and y coincides with the lefthand 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 onetoone correspondence within the class of submodular functions. It is not true, either, that the mapping f → f • gives a onetoone correspondence between the class of submodular functions and the class of supermodular functions.
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Chapter 8. Conjugacy and Duality
Polyhedral M/LConvex Functions
Conjugacy for polyhedral Mconvex and Lconvex functions is considered here. We start with a technical lemma. Proposition 8.3. Let g ∈ L[R → R] be a polyhedral Lconvex 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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which is a contradiction to p∗ ∈ arg min g[−x] and q∗ ∈ arg min g[−y]. The conjugacy theorem for polyhedral Mconvex and Lconvex functions is now stated. Theorem 8.4 (Conjugacy theorem). (1) The classes of polyhedral Mconvex functions and polyhedral Lconvex functions, M = M[R → R] and L = L[R → R], are in onetoone 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 onetoone 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 (MEXC [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 Mconvex polyhedra M0 [R] and positively homogeneous Lconvex functions 0 L[R → R] are conjugate to each other and also that Lconvex polyhedra L0 [R] and positively homogeneous Mconvex functions 0 M[R → R] are conjugate to each other. On the other hand, we can identify positively homogeneous Lconvex functions 0 L[R → R] with submodular set functions S[R] (Theorem 7.40) and positively homogeneous Mconvex 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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onetoone 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 Mconvex cones and Lconvex 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 Mconvex if and only if its polar cone is Lconvex. Hence, the classes of Mconvex cones and Lconvex cones are in onetoone correspondence under polarity (3.34). Taking integrality into account in diagram (8.9), we obtain
T [Z] ←→
M0 [ZR] M[ZR → R] M[R → RZ] 0 M[R → RZ]
←→ ←→ ←→ ←→
0 L[R
→ RZ] ←→ S[Z] L[R → RZ] L[ZR → R] L0 [ZR]
(8.10)
where M[ZR → R] and L[ZR → R] denote the sets of integral polyhedral Mconvex and Lconvex functions, respectively; M[R → RZ] and L[R → RZ] denote the sets of dualintegral polyhedral Mconvex and Lconvex functions, respectively; and 0 M[R → RZ] and 0 L[R → RZ] are their subclasses with positive homogeneity.52 It is known that the conjugacy relationship between Mconvexity and Lconvexity holds more generally for closed proper convex functions. Recall that the Legendre–Fenchel transformation gives a symmetric onetoone correspondence in the class of all closed proper convex functions (Theorem 3.2). Theorem 8.6. A closed proper convex function f satisﬁes (MEXC[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 (MEXC[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 → RZ] and L [R → RZ] denote the sets of dualintegral 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. (MEXC[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 Mconvex 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 Lconvex 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 Mconvex 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 Mconvex 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 Lconvex 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 Mconvex, 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 Lconvex cone in terms of a ring family D ⊆ 2V . Let D be an Lconvex cone and B be the polar of D. By Theorem 8.5, B is an Mconvex 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 Lconvex, as can be shown by reversing the above argument.
8.1.3
Integral M/LConvex 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 integervalued 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 Mconvex and Lconvex functions reads as follows. Theorem 8.12 (Discrete conjugacy theorem). (1) The classes of integervalued Mconvex functions and integervalued Lconvex functions, M = M[Z → Z] and L = L[Z → Z], are in onetoone 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 integervalued M convex functions and integervalued L convex functions, M [Z → Z] and L [Z → Z], are in onetoone 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 Loptimality criterion (Theorem 7.14 (2)), to α εχY , χv − χu ≤ g(p0 + εχY ) − g(p0 ) − εχY , x
(∀ Y ⊆ V, ε = ±1).
Note that the righthand 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 Mconvex and Lconvex functions are demonstrated below for integervalued 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

MSET = base polyhedron
MFNC
L2 FNC
 LPHF submod. set fnc. conjugate
MPHF distance fnc.

LFNC LSET
projection ?
restriction ?
M SET = gpolymatroid 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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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 positivedeﬁnite 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 Mconvex/Mconcave functions (Mseparation theorem), the separation theorem for Lconvex/Lconcave functions (Lseparation theorem), and the Fencheltype 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 Mconvex 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 Mconvex/Mconcave 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 (Mseparation 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 integervalued 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 integervalued Mconvex function is an Lconvex 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 Lconvex, convexity in intersection for Lconvex 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 Mconvex sets, the separation theorem for Mconvex sets (Theorem 4.21) gives p∗ ∈ RV such that the righthand 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 integervalued f and h, we have f • , −h◦ ∈ L[ZR → R] and, hence, domR f • , domR h◦ ∈ L0 [ZR]. 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 Lconvex/Lconcave functions. Theorem 8.16 (Lseparation 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 integervalued 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 integervalued Lconvex function is an Mconvex 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 Mconvex, convexity in intersection for Mconvex 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 Lconvex sets, the separation theorem for Lconvex sets (Theorem 5.9) gives x∗ ∈ RV such that the righthand 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 integervalued g and k we have g • , −k ◦ ∈ M[ZR → R] and, hence, domR g • , domR k ◦ ∈ M0 [ZR]. 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 Mseparation theorem we can obtain an optimality criterion for the problem of minimizing the sum of two Mconvex functions, which we call the Mconvex intersection problem. Note that the sum of Mconvex functions is no longer Mconvex and Theorem 6.26 (Moptimality criterion) does not apply. Theorem 8.17 (Mconvex 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 integervalued p∗ ∈ ZV . Proof. The suﬃciency of (8.25) and (8.26) is obvious. Conversely, suppose that (8.24) is true and apply the Mseparation 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 Moptimality 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 Mconvex function f : Z2 → Z ∪ {+∞} and an Mconcave 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 Mseparation theorem (Theorem 8.15). (A similar remark applies to the proof of the Lseparation theorem.) We applied the separation theorem to f and h, the convex and concave
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221
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 Mseparation theorem is based on an algorithmic argument for a generalization of the submodular ﬂow problem involving an Mconvex 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
FenchelType Duality Theorem
The Fencheltype 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/Mconcave functions and L convex/Lconcave functions and that all three inequalities are equalities for such integervalued functions.
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Theorem 8.21 (Fencheltype 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 integervalued M convex function and h : V Z → Z ∪ {−∞} be an integervalued 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 integervalued L convex function and k : V Z → Z ∪ {−∞} be an integervalued 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 Mseparation 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 Mconvex 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 Mseparation 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 Lconvex 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 Lseparation 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 Mseparation and Lseparation theorems are parallel or conjugate in their statements as well as in their proofs. In contrast, the Fencheltype duality theorem for integervalued functions is selfconjugate 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 Mseparation theorem and the Lseparation theorem independently and derived the Fencheltype duality theorem therefrom. It is noted, however, that, with the knowledge of M/Lconjugacy, 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 Mconvex and Mconcave 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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Mseparation theorem f (x) ≥ α∗ + p∗ , x ≥ h(x) Fencheltype duality (Fujishige [62]) ! Intersection theorem (Edmonds [44]) ⎧ ! ⎪ ⎪ ⎪ ⇒ Discrete separation for submodular functions Fencheltype duality ⎪ ⎨ (Frank [55]) inf{f − h} ⎪ ⎪ ⇒ Valuated matroid intersection = sup{h◦ − f • } ⎪ ⎪ ⎩ (Murota [135]) ! ⇓ Lseparation 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 Mconvex and Lconvex 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 Lseparation 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 Lseparation 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 Lseparation 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 Fencheltype duality theorem (Theorem 8.21 (3)). This is explained in Example 1.20. Example 8.26. The Fencheltype duality theorem for submodular set functions is a special case of the Fencheltype 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 Fencheltype 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 integervalued ρ and μ, the maximum on the righthand 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 weightsplitting theorem for the weighted matroid intersection problem is a special case of the optimality criterion for the Mconvex 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 weightsplitting 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 minimumweight base of (V, B1 ) with respect to w1∗ , and (iii) B ∗ is a minimumweight 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 integervalued weight. Applying Theorem 8.17 to a pair of Mconvex 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 weightsplitting theorem (integerweight case) from the Mseparation 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 weightsplitting 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 Mconvex 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 Mconvex 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 convexextensibility 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 Mconvex functions, and an L2 convex function is the integer inﬁmal convolution of two Lconvex functions. These functions play crucial roles in combinatorial optimization. In Edmonds’s intersection theorem (Theorem 4.18), for example, the lefthand side of the minmax relation (4.29) corresponds to M2 convexity and the righthand 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 Mconvex 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 Mconvex 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 polynomialtime 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 Mconvex 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 Mconvex 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 Mconvex 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 Mconvex 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 Mconvex 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 [ZR] 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 Mconvex functions as follows. Theorem 8.36. For integervalued 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 Mconvex 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 Lconvex 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 Lconvex 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 Lconvex 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 integervalued 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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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 Lconvex 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 Lproximity 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 [ZR], which is a variant of Theorem 7.43 (2). Theorem 8.46. For integervalued 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 Lconvex 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 onetoone 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 Mconvex set, and (OBJ) c is an Mconvex function. We refer to a problem with (REG) and (OBJ) as an Mconvex 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 Fencheltype duality theorem for M/Lconvex 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 perturbationbased 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 Mconvex programs as the wellbehaved 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 integervalued 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 (Saddlepoint 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 MConvexity
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 Mconvex 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/Lconvex functions, whereas the general case involves M2 /L2 convex functions. Throughout this section we assume (REG), i.e., that B is an Mconvex 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 Mconvex 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 Mconvex set. Therefore, δB (x + u) is Mconvex in u. (3) We have Fr (x, u) = +∞ unless x ∈ dom c. Besides δB (x + u), r(u) is Mconvex by the assumption. Hence, Fr (x, ·) is the sum of two Mconvex 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 Lconcave 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 Mconvex 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 Mconvex or ∈ {+∞, −∞} since it is the integer inﬁmal convolution of two Mconvex functions (Theorem 6.13 (8)). In the case of Mconvex 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 Lconcave 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 Mconvex 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 Lconcave or ≡ −∞, since the sum of two Lconcave functions is again Lconcave 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 Mconvex 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 Mconvexity 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 Mconvex 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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Chapter 8. Conjugacy and Duality
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 Mconvex 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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8.4. Lagrange Duality for Optimization
243
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 Mconvex. 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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244
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 Mconvexity and Lconvexity was established ﬁrst for integervalued functions (Theorem 8.12) by Murota [140], whereas the present proof is based on Murota [147]. The conjugacy theorem for polyhedral M/Lconvex functions (Theorem 8.4) is due to Murota–Shioura [152]. The polarity between M/Lconvex 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 Mseparation theorem (Theorem 8.15) is given in Murota [137], [140], [142] and the Lseparation theorem (Theorem 8.16) in [140]. The Fencheltype duality theorem for Mconvex functions originated in [137] (see also [140]); the present form (Theorem 8.21) is in Murota [147]. The Mconvex intersection theorem (Theorem 8.17) is in [137], [142]. The Fencheltype duality theorem for submodular set functions described in Example 8.26 is due to Fujishige [62]. The weightsplitting 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/Lconvexity 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/Lconvexity plays a fundamental role. (ii) The submodular ﬂow problem with an Mconvex function admits nice optimality criteria in terms of potentials and negative cycles. (iii) The optimality criterion using potentials is equivalent to the Fencheltype duality theorem. (iv) A conjugate pair of Mconvex and Lconvex functions is transformed to another conjugate pair of Mconvex and Lconvex 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/Lconvexity 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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Chapter 9. Network Flows
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 wellbehaved 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 integervalued functions, c : A → Z ∪ {+∞}, c : A → Z ∪ {−∞}, and x : V → Z, then there exists an integervalued optimal ﬂow ξ to the above problem. This implies that the problem MCFP0 speciﬁed by such integervalued 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 integerﬂow 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 integerﬂow version of the problem it is natural to assume fa ∈ C[Z → R] (or fa ∈ C[ZR → R]) for each a ∈ A, but it is not clear what combinatorial property to impose on f to ensure the integrality of optimal solutions. Mconvexity 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/Lconvex 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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Chapter 9. Network Flows
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 maxﬂow mincut 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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249
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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250
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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252
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 (Negativecycle 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 integervalued optimal ﬂow exists if the upper and lower capacities and the supply vector are integer valued (primal integrality). 2. An integervalued 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 integervalued optimal ﬂow ξ : A → Z. (2) [Dual integrality] The set of optimal potentials Π∗ = {p  p : optimal potential } is an Lconvex polyhedron. If γ : A → Z, then Π∗ is an integral Lconvex polyhedron and there exists an integervalued 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 Lconvexity 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 Mconvexity 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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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 minmax 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 lefthand side of (9.36) is MCFP3 in disguise, and accordingly, we may think of the maximization problem on the righthand 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 = ∂ξV1 and p = p1 . This implies the Fenchel duality (3.41) for f1 and h2 ; see also (3.30) and (3.42).
9.2
MConvex Submodular Flow Problem
A series of generalizations of the minimum cost ﬂow problem to the Mconvex 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 Mconvex 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 integerﬂow 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 Mconvex function. Such a problem described by some f ∈ M[R → R] is called the Mconvex submodular ﬂow problem. Mconvex 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 Mconvex submodular ﬂow problem with a {0, +∞}valued f reduces to the submodular ﬂow problem MSFP1 . In the integerﬂow version of the problem we assume c : A → Z ∪ {+∞}, c : A → Z ∪ {−∞}, and f ∈ M[Z → R] (or f ∈ M[ZR → 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 Mconvex. Mconvex 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 integerﬂow version of the problem we assume f ∈ M[Z → R] and fa ∈ C[Z → R] for a ∈ A (or f ∈ M[ZR → R] and fa ∈ C[ZR → 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 Mconvex 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 (˜ xV ) 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 Mconvex 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 integerﬂow version of such a problem, however, contains the Hamiltonian path problem, a wellknown NPcomplete 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 integervalued 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 integervalued c, c, and ρ, both B(κ) and B(ρ) are integral base polyhedra (integral Mconvex 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 maxﬂow mincut 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 (Maxﬂow mincut 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 integervalued 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 Mconvex 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 Mconvex 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 Mconvexity 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 Mconvex, 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 Moptimality 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 cyclecanceling algorithm for the Mconvex submodular ﬂow problem, to be explained in section 10.4.3.
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An alternative representation of condition (ii) in (POT) is obtained from the Mconvexity of f . The function conjugate to f , say, g, is a polyhedral Lconvex function with g(p + 1) = g(p) for all p (by Theorem 8.4 and (9.51)). It follows from (3.30) and the Loptimality 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 primaldual algorithm for the Mconvex submodular ﬂow problem, to be explained in section 10.4.4. We go on to discuss integrality properties of the Mconvex submodular ﬂow problem. This generalizes the wellknown facts (Theorem 9.6) for the minimum cost ﬂow problem MCFP0 . Recall the notation M[ZR → R] and M[R → RZ] for the sets of integral and dualintegral polyhedral Mconvex functions, respectively. Theorem 9.15. Suppose that an optimal solution exists in the Mconvex 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 Lconvex polyhedron. (2) [Primal integrality] If fa ∈ C[ZR → R] (a ∈ A) and f ∈ M[ZR → R], then ∂Ξ∗ is an integral M2 convex polyhedron, and there exists an integervalued optimal ﬂow ξ : A → Z. (3) [Dual integrality] If fa ∈ C[R → RZ] (a ∈ A) and f ∈ M[R → RZ], then Π∗ is an integral Lconvex polyhedron, and there exists an integervalued 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 Lconvex polyhedron, say, D1 , by (9.63), whereas those satisfying (ii) in (POT) form another Lconvex polyhedron D2 by (9.64). Therefore, Π∗ = D1 ∩ D2 is an Lconvex polyhedron. (2) Both B(κ∗ ) and B(gp ) are integral Mconvex polyhedra. The integrality ∗ of ∂Ξ = B(κ∗ ) ∩ B(gp ) follows from (4.32). (3) Both D1 and D2 are integral Lconvex 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[ZR → R] ⇐⇒ c(a), c(a) ∈ Z,
(9.66)
fa ∈ C[R → RZ] ⇐⇒ γ(a) ∈ Z.
(9.67)
Finally, we state the optimality criterion for the integerﬂow version of the Mconvex submodular ﬂow problem MSFP3 . This is a corollary of Theorems 9.14 and 9.15. Theorem 9.16 (Potential criterion). Consider the Mconvex 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 integervalued potential p : V → Z in (POT). Moreover, the set of integervalued optimal potentials, Π∗ = {p  p : integervalued optimal potential}, is an Lconvex 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 Fencheltype duality theorem for Mconvex functions (Theorem 8.21) is essentially equivalent to the optimality criterion for the Mconvex submodular integerﬂow problem (Theorem 9.16). See section 9.1.4 and note that, for an Mconvex function f1 and an Mconcave function h2 , f (x1 , x2 ) = f1 (x1 ) − h2 (−x2 ) is an Mconvex function.
9.5
Optimality Criterion by Negative Cycles
The optimality of an Mconvex submodular ﬂow can also be characterized by the nonexistence of negative cycles in an auxiliary network. This fact leads to the cyclecanceling algorithm to be described in section 10.4.3.
9.5.1
NegativeCycle Criterion
We consider the Mconvex submodular ﬂow problem MSFP2 with Mconvex 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 realvalued ﬂows and then integervalued ﬂows. We assume f ∈ M[R → R] in considering realvalued 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 (Negativecycle criterion). For a feasible ﬂow ξ : A → R to the Mconvex 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 integervalued ξ , we can take an integervalued p in the proof of Theorem 9.18. Next we consider the integerﬂow 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 (Negativecycle criterion). For a feasible integer ﬂow ξ : A → Z to the Mconvex 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 Mconvex intersection problem introduced in section 8.2.1 can be formulated as an Mconvex submodular ﬂow problem. Given two Mconvex functions f1 , f2 : ZV → R ∪ {+∞}, we consider an Mconvex 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 Mconvex intersection problem.
identically zero without capacity constraints. Since x1 = −x2 if (x1 , x2 ) = ∂ξ for a ﬂow ξ in this network, the Mconvex submodular ﬂow problem is equivalent to minimizing f1 (x) + f2 (x). The negativecycle optimality criterion (Theorem 9.20) for this Mconvex 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 negativecycle 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 integerﬂow 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 Mconvex 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 uniquemin 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 uniquemin condition if there exists in G(x, y) exactly one minimumweight 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 uniquemin condition is a suﬃcient condition for this inequality to be an equality. Proposition 9.23. Let f ∈ M[Z → R] be an Mconvex function, and assume x ∈ domZ f , y ∈ ZV , and x−y∞ = 1. If (x, y) satisﬁes the uniquemin 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 uniquemax 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 minimumweight perfect matching. It also shows that the uniquemin 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 minimumweight 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 uniquemin 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 − ∂ξ1 /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 minimumweight 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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Chapter 9. Network Flows
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 Mconvex and Lconvex functions. A given pair of Mconvex and Lconvex functions deﬁned on entrance vertices of a network is transformed through the network to another pair of Mconvex and Lconvex 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/Lconvexity 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, integervalued 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 integervalued 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/Lconvexity by which a conjugate pair of Mconvex and Lconvex functions is transformed to another conjugate pair of Mconvex and Lconvex functions. Note that C[Z → Z] denotes the set of univariate integervalued 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 wellknown 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 vertexdisjoint 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 Mconvexity 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 minmax 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 integervalued 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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272
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 Mconvex and Lconvex functions can be formulated as the transformation by networks, as is partly demonstrated below.
Note 9.29. The Mconvexity of the aggregation f U∗ of an Mconvex 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 Mconvex submodular ﬂow problem. Note 9.30. The Mconvexity of the inﬁmal convolution f1 2Z f2 of Mconvex 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 Mconvex submodular ﬂow problem.
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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 (MEXC[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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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 (MEXC[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 (bS ) +
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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with bS 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 ﬂowboundary 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 ﬂowboundary tuple (ϕ1 , ϕ2 , b1 , b2 , w) is updated to another ﬂowboundary 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 (MEXC[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 ﬂowboundary tuple (ϕ◦1 , ϕ◦2 , b◦1 , b◦2 , w◦ ) with the same frontier vertex w◦ = w. The ﬂowboundary 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 ﬂowboundary tuples. The generation of ﬂowboundary 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 ﬂowboundary 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 ﬂowboundary 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 ﬂowboundary 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 ﬂowboundary tuples. This is because (i) the frontier vertices in one stage are distinct and hence the number of ﬂowboundary 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 ﬂowboundary 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 crossingsubmodular functions. The present form avoids crossingsubmodular functions on the basis of the fact, due to Fujishige [61], that the base polyhedron deﬁned by a crossingsubmodular 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 Mconvex 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 Mconvex submodular ﬂow problem was established by Murota [142] for the integerﬂow version (Theorem 9.16) and adapted to the realvalued 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 integerﬂow version and adapted to the realvalued case (Theorem 9.18) in Murota–Shioura [152]. Theorem 9.22 is in [140], [142]. Proposition 9.23 is a reformulation of the uniquemax 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 Mconvex functions f ∈ M[Z → R] by Murota [137]; the proof given in section 9.6.2 is due to Shioura [188], [189]. Transformation of Lconvex functions is stated explicitly in Murota [145]. The extension to polyhedral Mconvex and Lconvex 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 Mconvex and Lconvex functions are discussed in this chapter. Three fundamental optimization problems tractable by eﬃcient algorithms are (i) Mconvex function minimization, which is a nonlinear extension of the minimumweight base problem for matroids; (ii) Lconvex function minimization, which includes submodular set function minimization as a special case; and (iii) minimization/maximization in the Fencheltype minmax duality, which is equivalent to the Mconvex submodular ﬂow problem.
10.1
Minimization of MConvex Functions
Four kinds of algorithms for Mconvex 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 Mconvex 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 Mconvex functions (Theorem 6.26) immediately suggests the following algorithm of steepest descent type. Steepest descent algorithm for an Mconvex 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 (Moptimality 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 (Mminimizer 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 Mconvex 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 − y1  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 Mconvex 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 tiebreaking 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 tiebreaking rule we have the following complexity bound. Proposition 10.2. For an Mconvex function f with ﬁnite K1 in (10.1), the number of iterations in the steepest descent algorithm with tiebreaking 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 Moptimality criterion) and Theorem 6.77 (quasi Mminimizer cut), the steepest descent algorithm can also be used for minimizing quasi Mconvex 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 Mconvex 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 Mconvex functions leads us to the following steepest descent scaling algorithm for Mconvex 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 Mconvex 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 Mproximity 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 Moptimality 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 Mconvex, step S1 can be done in O(F · n4 ) time by the steepest descent algorithm with tiebreaking rule (10.2). This time bound follows from Proposition 10.2 with K1 ≤ 4n2 . For a general f , however, f˜ is not necessarily Mconvex (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 Moptimality criterion) and Theorem 6.78 (quasi Mproximity theorem), the steepest descent scaling algorithm can be adapted to the minimization of quasi Mconvex 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 Mconvex function by generating a sequence of nested subsets of the domain on the basis of the Mminimizer cut theorem (Theorem 6.28). For an Mconvex 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 Mconvex; see Proposition 10.6 below. Domain reduction algorithm for an Mconvex 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 Mconvex sets, which contain a minimizer of f because of the Mminimizer 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 Mconvex 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 Mconvex 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 Mconvex 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 (MEXC[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}, (MEXC[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 Mconvex 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 Mminimizer cut), the domain reduction algorithm can also be used for minimizing quasi Mconvex functions satisfying (SSQM = ) provided that the eﬀective domain is a bounded Mconvex set.
10.1.4
Domain Reduction Scaling Algorithm
We present here the domain reduction scaling algorithm for Mconvex 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 Mminimizer 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 αscaling phase is described later and K∞ is deﬁned in (10.3).
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Domain reduction scaling algorithm for an Mconvex 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) (αscaling phase). S2: If α = 1, then stop (x is a minimizer of f ). S3: Set α := α/2 and go to S1. The αscaling 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. αscaling 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 αscaling phase is bounded by x◦ − ◦ 1 /α. In particular, the αscaling 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 Mconvex function f in O(F · n3 log2 (K∞ /n)) time. Proof. At the beginning of the αscaling phase we have x◦ −◦ 1 ≤ n(n−1)(2α−1) by (10.6). Since step S3 in the αscaling phase can be done with n evaluations of f , the αscaling 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 Mminimizer cut with scaling), the domain reduction scaling algorithm can be adapted to the minimization of quasi Mconvex functions satisfying (SSQM = ) provided that the eﬀective domain is a bounded Mconvex 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 Lconvex 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 lefthand 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 minmax 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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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 mincut 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) + εX 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 ﬁnitevalued 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 tcomponent 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 integervalued 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 δfeasible 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 δfeasible} 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 δfeasible 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 δfeasible 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 integervalued ρ. In the scaling phase with parameter value δ we maintain a δfeasible 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 δaugmenting path. We also maintain a base x represented in the form of (10.13) with {(Li , λi )  i ∈ I}. If there exists a δaugmenting path P , we modify ϕ to another δfeasible 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 δaugmenting 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 minmax 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 ) − δW  − δV \ 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 δfeasible. 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 DoubleExchange(i, u, v) below. Procedure DoubleExchange(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 integervalued 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 δaugmenting path, apply Augment(ϕ, P ) and Reduce(x, I), and go to S1. S3: If there exists an active triple, then apply DoubleExchange 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 DoubleExchange. 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 integervalued 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 DoubleExchange. (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 DoubleExchange 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 DoubleExchange. 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 δaugmenting 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 DoubleExchange 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 ﬁnitevalued 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 minmax 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 LConvex Functions
Three kinds of algorithms for Lconvex 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 Lconvex 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 Lconvex functions (Theorem 7.14) naturally leads to the following steepest descent algorithm. Steepest descent algorithm for an Lconvex 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) (Loptimality 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 tiebreaking 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 tiebreaking 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 − q1  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 Lconvex function g with ﬁnite K iterations in the steepest descent algorithm with tiebreaking 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 Lconvex function g˜ as (p0 ∈ Z, p ∈ ZV ). (10.35) g˜(p0 , p) = g(p − p0 1) The steepest descent algorithm above applied to this Lconvex 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 tiebreaking 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 tiebreaking 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) (Loptimality 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 − q1  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 − q1 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− q1 = p − q 1 ≤ K1 . In case (ii), we have p0 − q0 = q (v)− p (v) and ˆ 1 (˜ K g ) = p0 − q0  + p − q1 = (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 Lconvex functions satisfying (SSQSBw ) because of Theorem 7.53 (quasi Loptimality 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 Lconvex 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 Lconvex functions. The algorithm for an Lconvex 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 Lconvex 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 Lconvex function. By the Lproximity 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 tiebreaking 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 (Loptimality 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 Lconvex functions satisfying (SSQSB) because of Theorem 7.53 (quasi Loptimality criterion) and Theorem 7.54 (quasi Lproximity 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 MConvex Submodular Flows
Five algorithms for the Mconvex submodular ﬂow problem are described: the twostage algorithm, the successive shortest path algorithm, the cyclecanceling algo
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rithm, the primaldual algorithm, and the conjugate scaling algorithm. Because the optimality criterion for the Mconvex submodular ﬂow problem is essentially equivalent to the duality theorems for M/Lconvex 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 Fencheltype duality.
10.4.1
TwoStage Algorithm
This section is intended to provide a general structural view on the duality nature of the Mconvex submodular ﬂow problem. It is based on the recognition of the Mconvex submodular ﬂow problem as a composition of the Fencheltype duality and the minimum cost ﬂow problem that does not involve an Mconvex function. The algorithm presented in this section, called the twostage algorithm, computes an optimal potential by solving an Lconvex 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 Mconvex submodular ﬂow problem MSFP3 and the Fencheltype duality may be summarized as follows. To be speciﬁc, we consider the integerﬂow version of MSFP3 on the graph G = (V, A) with f ∈ M[Z → Z] and fa ∈ C[Z → Z] for a ∈ A. Mconvex 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 Fencheltype 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 righthand side of (10.44) is an Lconvex 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 Mconvex 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 twostage algorithm is described as follows. Twostage 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 Mconvex 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 Lconvex function minimization algorithm to g˜. 3. We can ﬁnd a member of B ∗ by applying an Mconvex 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 Mconvex 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 − ∂ξ1 = 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 − ∂ξ1 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 uniquemin 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 uniquemin 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 realvalued 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 integerﬂow version of MSFP2 (with Mconvex 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
CycleCanceling Algorithm
For the Mconvex submodular integerﬂow 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 cyclecanceling algorithm, which works on the auxiliary network (Gξ , ξ ) introduced in section 9.5. Cyclecanceling 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 cyclecanceling algorithm are known to be strongly polynomial (see Murota [136]). Cyclecanceling algorithms can also be designed for problems with realvalued ﬂow on the basis of Theorem 9.18. Note 10.36. For MSFP1 (submodular ﬂow problem with linear arc cost and without Mconvex cost), a number of cyclecanceling algorithms are proposed, including Fujishige [59], Cui–Fujishige [27], Zimmermann [222], Wallacher–Zimmermann [210], and Iwata–McCormick–Shigeno [103].
10.4.4
PrimalDual Algorithm
A primaldual algorithm for the Mconvex 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. Mconvex 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 → RZ], 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 integervalued 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 (integervalued) 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[ZR → 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 (Mconvex polyhedron) associated with gp . The algorithm maintains a pair (ξ, p) ∈ RA × ZV of a feasible ﬂow ξ and an integervalued 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 outofkilter 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 outofkilter 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 outofkilter 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 primaldual algorithm picks an outofkilter arc and tries to meet (10.57) and (10.58) for that arc. The present strategy to pick a vertex is to improve the worstcase 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 primaldual algorithm for the Mconvex submodular ﬂow problem with a linear arc cost (MSFP2 ) with dual integrality is summarized as follows. Primaldual 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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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 (maxﬂow mincut 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 primaldual algorithm. Proposition 10.39. (1) The set of outofkilter 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 primaldual 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 inkilter 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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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 ) = ∅. Primaldual algorithms can also be designed for problems without dual integrality by using realvalued potential functions. Note 10.40. The framework of the primaldual algorithm for MSFP1 (submodular ﬂow problem without Mconvex 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 Mconvex cost function, the primaldual algorithm is enhanced to a polynomialtime algorithm. We continue to deal with the Mconvex submodular ﬂow problem MSFP2 with dual integrality; i.e., we assume γ : A → Z and f ∈ M[R → RZ]. First we explain the intuition behind the costscaling algorithm for the submodular ﬂow problem MSFP1 without Mconvex function (see section 9.2 for MSFP1 ). As Proposition 10.39 (4) shows, the time complexity of the primaldual algorithm depends essentially on γp (a) = γ(a) + p(∂ + a) − p(∂ − a).
(10.74)
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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 αγ(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 primaldual algorithm. The scaling algorithm embodies the above idea by starting with a large α and successively halving α until α = 1. When an Mconvex 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 Mconvex for an Mconvex function f . Conjugate scaling is a kind of scaling operation compatible with Mconvexity. 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 dualintegral 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 dualintegral polyhedral Mconvex function f ∈ M[R → RZ], we have f α ∈ M[R → RZ] provided f α > −∞. Proof. We have f • = g ∈ L[ZR → R] and hence g α ∈ L[Z → R] by Theorem 7.10 (2). Therefore, f α = (g α )• ∈ M[R → RZ] 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 integervalued 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 → RZ] 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 integervalued 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 primaldual 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 (Lproximity 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 Lconvex 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 Mconvex 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[ZR → 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 primaldual 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 outofkilter 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 righthand side we consider only those x, u, v for which f (x; −χu + χv ) is ﬁnite. If f ∈ M[ZR → 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 tiebreaking rule (10.2) for Mconvex 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 Mconvex 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 Mconvex functions that are closed under scaling. A polynomialtime scaling algorithm for general Mconvex 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 Mconvex functions is observed in Murota– Shioura [154]. The domain reduction scaling algorithm in section 10.1.4, with its extension to quasi Mconvex 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 Lconvex function minimization, when translated through the equivalence between submodular integrally convex functions and L convex functions. The steepest descent algorithm for Lconvex function minimization in section 10.3.1 is given in Murota [145]. The tiebreaking 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 Mconvex 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 cyclecanceling algorithm of section 10.4.3 is devised in [142] as a proof of the negativecycle 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 primaldual algorithm of section 10.4.4 is due to Iwata–Shigeno [105]; see also Note 10.40. A strongly polynomial primaldual 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 Mconvex submodular ﬂow problem is given by Moriguchi–Murota [132]. Capacity scaling algorithms for submodular ﬂow problems (without Mconvex costs) are given in Iwata [97] and Fleischer–Iwata–McCormick [51]. For other algorithms for submodular ﬂow problems (without Mconvex 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 realvalued vectors, a rigorous mathematical framework was established around 1960 on the basis of convexity, compactness, and ﬁxedpoint 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 Mconvexity and Lconvexity 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 integervalued 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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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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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 supplydemand 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 righthand 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 economictheoretical 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 Mconvex 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 realvalued 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 ﬁxedpoint 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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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 singlesource 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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345
The computation of an equilibrium via an Mconvex submodular ﬂow problem described in section 11.5 is due to Murota–Tamura [161]. Mconvexity 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 graphtheoretic 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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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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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 sixdimensional vector x in our description of the system. It is possible, however, to describe this system using a fourdimensional 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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12.2. Mixed Matrices and Mixed Polynomial Matrices
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 (MQ) Q = (Qij ) is a matrix over K and (MT) 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 (MPQ) Qk (k = 0, 1, . . . , N ) are matrices over K and (MPT) 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 timeinvariant dynamical systems. The variable s here is primarily intended to denote the variable for the Laplace transform for continuoustime systems, though it could be interpreted as the variable for the ztransform 68 for discretetime 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 Qpart numerically and the T part symbolically. This is based on the following observations. 68 See
Chen [23] or Zadeh–Desoer [220] for the ztransform.
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Qpart: As is typical with electrical networks, the Qpart primarily represents the interconnection of the elements. The Qmatrix, 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 Qpart should be treated numerically or linear algebraically. In fact, this is feasible in practice, since the entries of the Qmatrix 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 graphtheoretic concepts and algorithms. Combination: The structural information from the Qpart 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: Qpart 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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12.3. Rank of Mixed Matrices
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 righthand 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 righthand side of the identity (12.9) is a maximization over all pairs (I, J), the number of which is as large as 2R+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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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 minmax 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 righthand 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 righthand 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 righthand 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 Mconvex 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 Mconvex 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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360
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 Mconcave, 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 righthand side of the identity (12.14) is a maximization over all pairs (I, J), the number of which is as large as 2R+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 Mconvex submodular ﬂow problem. To see the connection to Mconvexity, 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 Mconvex functions. The righthand side of (12.14) can now be identiﬁed as the negative of an integer inﬁmal convolution of these Mconvex 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 Mconvex functions can be computed eﬃciently by solving an Mconvex 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 Mconvexity 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 skewsymmetric 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 skewsymmetric 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 cyclecanceling, 313 domain reduction, 284 domain reduction scaling, 287 fully combinatorial, 290 greedy, 3, 108 IFF ﬁxing, 300 IFF scaling, 299 Lconvex function minimization, 305, 306, 308 Mconvex function minimization, 281, 283, 284, 287 primaldual, 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 twostage, 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 quasiseparable, 334 separable, 333 conductance, 41 cone, 78 convex, 78 Lconvex, 131 Mconvex, 119 polar, 82 conformal decomposition, 64 conjugacy theorem closed proper M/Lconvex, 210 in convex analysis, 11, 82 discrete M/Lconvex, 30, 212 polyhedral M/Lconvex, 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 dualintegral polyhedral, 161 integral polyhedral, 161 laminar, 141 polyhedral, 80 positively homogeneous, 82 proper, 77 quadratic, 40 quasi, 168 quasiseparable, 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
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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 cyclecanceling 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 Lconvex function, 218 Lconvex set, 36, 126 Mconvex function, 217 Mconvex set, 36, 114 submodular function, 17, 111 submodular function (as special case of Lseparation), 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 dualintegral polyhedral convex function, 161 Lconvex function, 191 Mconvex function, 161 dual integrality intersection theorem, 20, 114 linear programming, 89 minimum cost ﬂow problem, 252 polyhedral convex function, 161 polyhedral Lconvex function, 191 polyhedral Mconvex 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 Fencheltype, 222, 225 Lseparation, 218 linear programming, 87 Mseparation, 217 matroid intersection, 225 separation for convex functions, 84 separation for convex sets, 35, 83 separation for Lconvex functions, 218 separation for Mconvex 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
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384 (as special case of Fencheltype 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 (BEXC[R]), 118 (BEXC+ [R]), 118 (BEXC[Z]), 18, 101 (BEXC+ [Z]), 102 (BEXC− [Z]), 103 (BEXCw [Z]), 103 (B EXC[R]), 118 (B EXC[Z]), 117 local, 135 Mconvex function, 26, 58, 133 Mconvex polyhedron, 118 Mconvex set, 18, 101 M convex function, 27, 134 M convex polyhedron, 118 M convex set, 117 (MEXC[R]), 29, 56, 160 (MEXC [R]), 160 (MEXC[Z]), 26, 58, 133 (MEXC [Z]), 26, 133 (MEXCloc [Z]), 135 (MEXCw [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 Mconvex 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 Fencheltype duality generic form, 13 Lconvex function, 32, 222 Mconvex function, 32, 222 submodular function, 225 ﬁxed constant, 347 ﬂow, 53 δfeasible, 296 feasible, 247, 258 Frank’s discrete separation theorem, 17, 111 (as special case of Lseparation), 33, 224 Frank’s weightsplitting theorem, 34, 225 fully combinatorial algorithm, 290
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Index fundamental circuit, 149 gpolymatroid, 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 Lconvex polyhedron, 131 integral Mconvex polyhedron, 118 integral neighborhood, 93 integral polyhedral convex function, 161 Lconvex function, 191 L convex function, 192 Mconvex 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 Lconvex function, 191 polyhedral Mconvex 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 Mconvex, 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 Mconvex, 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 Lconcave function, 22 polyhedral, 190 Lconvex cone, 131 Lconvex function, 8, 22, 177 dualintegral polyhedral, 191 integral polyhedral, 191 polyhedral, 190 positively homogeneous, 193 quadratic, 52, 182 quasi, 199 semistrictly quasi, 199 Lconvex polyhedron, 123, 131 integral, 131 Lconvex set, 22, 121 Loptimality criterion, 185, 193 quasi, 201 Lproximity theorem, 186 quasi, 201 Lseparation 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 Mconcave function, 8, 26 polyhedral, 160 Mconvex cone, 119 Mconvex function, 8, 26, 133 dualintegral polyhedral, 161
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Index integral polyhedral, 161 polyhedral, 160 positively homogeneous, 164 quadratic, 52, 139 quasi, 169 semistrictly quasi, 169 Mconvex intersection problem, 219, 264 theorem, 219 Mconvex polyhedron, 108, 118 integral, 118 Mconvex program, 235 Mconvex set, 27, 101 Mconvex submodular ﬂow problem, 256 economic equilibrium, 341 Mmatrix, 42 Mminimizer cut, 149 with scaling, 158 Moptimality criterion, 148, 163 quasi, 173 Mproximity theorem, 156 quasi, 174 Mseparation 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 skewsymmetric, 361 node admittance, 42 polynomial, 71, 354 positivedeﬁnite, 39 positivesemideﬁnite, 39 principal sub, 40 totally unimodular, 88 matroid, 70 induction through a graph, 270 intersection problem, 34, 225 valuated, 72 maxﬂow mincut 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 minmax relation, 2 minimizer, 79 global, 9, 79 integrally convex function, 94 Lconvex function, 185, 305 L2 convex function, 232 local, 10 Mconvex function, 148, 281 M2 convex function, 227, 228 maximal, 290, 291, 304, 307 minimal, 290, 291, 305, 307 submodular set function, 288 minimizer cut Mconvex function, 149 Mconvex function with scaling, 158 quasi Mconvex function, 174 quasi Mconvex 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 skewsymmetric 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 Lconvexity, 24, 31, 56, 58, 270 Mconvexity, 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 oﬀdiagonal 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 Lconvex function, 185, 193 L2 convex function, 232 Mconvex function, 148, 163 Mconvex 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 Lconvex function, 201 quasi Mconvex function, 173 submodular ﬂow, 260 submodular set function, 185 sum of Mconvex 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 Lconcave function, 190 Lconvex function, 190 L convex function, 192 Mconcave function, 160
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Index Mconvex function, 160 M convex function, 161 method, 8 polyhedron base, 105 convex, 78 integral, 90 integral Lconvex, 131 integral Mconvex, 118 Lconvex, 123, 131 L convex, 129, 131 Mconvex, 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 Lconvex function, 193 Mconvex function, 164 potential, 41, 53, 89, 248 criterion, 249, 260, 262 optimal, 251 primaldual 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 Mconvex function, 134 Mconvex polyhedron, 118 Mconvex set, 102
389 M2 convex function, 226 M2 convex set, 117 polyhedral Mconvex function, 161 proper convex function, 77 proximity theorem, 156 Lconvex function, 186 L2 convex function, 232 Mconvex function, 156 M2 convex function, 228 quasi Lconvex function, 201 quasi Mconvex function, 174 pseudopolynomial algorithm, 288 quadratic form, 39 function, 39 Lconvex function, 182 L convex function, 48, 52, 182 Mconvex function, 139 M convex function, 48, 52, 139 quasi convex, 168 semistrictly, 168 quasi Lconvex function, 199 quasi Loptimality criterion, 201 quasi Lproximity theorem, 201 quasi linear, 324 quasi Mconvex function, 169 quasi Mminimizer cut, 174 with scaling, 175 quasi Moptimality criterion, 173 quasi Mproximity theorem, 174 quasiseparable 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 Lconvex function, 178 Lconvex polyhedron, 131 Lconvex set, 121 L2 convex function, 229 L2 convex set, 129 polyhedral Lconvex function, 192 rigidity, 361 ring family, 104, 107, 292 saddlepoint theorem, 238 scaling, 145 conjugate, 319 cost, 318 domain, 145 nonlinear, 170, 199 scaling algorithm Lconvex function, 308 Mconvex function, 283, 287 Mconvex submodular ﬂow, 320 semigroup, 45 semistrictly quasi convex, 168 semistrictly quasi Lconvex, 199 semistrictly quasi Mconvex, 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 Lconvex function, 218 Lconvex set, 36, 126 Mconvex function, 217 Mconvex 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 Lconvex function, 305, 306 Mconvex function, 281 steepest descent scaling algorithm Lconvex function, 308 Mconvex 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 Mconvex, 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 Mconvex 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 twostage algorithm, 310 unimodular totally, 88 uniquemin 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 ztransform, 355