Computing and Combinatorics: 17th Annual International Conference, COCOON 2011, Dallas, TX, USA, August 14-16, 2011. Proceedings (Lecture Notes in Computer Science, 6842) 3642226841, 9783642226847

Table of contents :
Title
Preface
Organization
Table of Contents
Derandomizing HSSW Algorithm for 3-SAT
Introduction
Preliminaries
A Derandomization of HSSW
Conclusion
References
Dominating Set Counting in Graph Classes
Introduction
Interval Graphs
Counting the Dominating Sets
Speeding Up the Dynamic Programming
Counting the Minimum Dominating Sets
Split Graphs
Cobipartite Graphs
Chordal Bipartite Graphs
Concluding Remarks
References
The Density Maximization Problem in Graphs
Introduction
Bi-constrained Maximum Density Subgraph
An FPTAS for Relaxed Density Maximization
Conclusion and Outlook
References
FlipCut Supertrees: Towards Matrix Representation Accuracy in Polynomial Time
Introduction
Preliminaries
The FlipCut Algorithm
Using Branch Lengths
The Undisputed Sibling Problem
Experiments
Results
Conclusion
References
Tight Bounds on Local Search to Approximate the Maximum Satisfiability Problems
Introduction
Local Search for Max-(2)-Sat
The 4/3-Approximation Algorithm for Max-E2-Sat
On the General Case of Max-(k)-Sat
The 8/7-Approximation Algorithm for Max-(3)-Sat
The 8/7-Approximation Algorithm for Max-E3-Sat
On the General Case of Max-(k)-Sat
Local Search for Other Variants
Conclusion
References
Parameterized Complexity in Multiple-Interval Graphs: Partition, Separation, Irredundancy
Introduction
Preliminaries
Vertex Clique Partition
Separating Vertices
Irredundant Set
References
Exact Parameterized Multilinear Monomial Counting via k-Layer Subset Convolution and k-Disjoint Sum
Introduction
Related Work
Preliminaries
Zeta Transform and Möbius Inversion
Subset Convolution
Faster Algorithms for Exact Multilinear k-Monomial Counting
k-Disjoint Sum
Algorithms for Exact Multilinear k-Monomial Counting
A Polynomial Space Algorithm for Exact Multilinear k-Monomial Counting
Applications
#k-Path
#m-Set k-Packing
Conclusions
References
On the Rainbow Connectivity of Graphs: Complexity and FPT Algorithms
Introduction
Complexity for Edge-Coloring Version
Strong NP-Completeness for Outerplanar Graphs
Polynomial-Time Algorithm for Cacti
Complexity for Variants
FPT Algorithms
FPT Algorithm for Edge-Coloring Version
FPT Algorithms for Two Variants
References
On Parameterized Independent Feedback Vertex Set
Introduction
An O^*(5^k) Algorithm
A Cubic Kernel
Discussion and Conclusion
References
Cograph Editing: Complexity and Parameterized Algorithms
Introduction
Preliminaries
Notations
Spider Graphs
P_4-Sparse Graphs
NP-Hardness
A Parameterized Algorithm
Editing P_4-Sparse Graphs to Cographs
Editing Graphs to P_4-Sparse Graphs
Conclusions
References
Approximation Complexity of Complex-Weighted Degree-Two Counting Constraint Satisfaction Problems
Bounded-Degree #CSPs
Fundamental Notions and Notations
Signatures, #CSPs, and Holant Problems
FPC and AP-Reducibility
Main Theorems
Symmetric Signatures of Arity 3
Arbitrary Signatures of Arity 3
T_2-Constructibility Technique
Parametrized Symmetrization Technique
Parametrized Symmetrization Scheme
Proof of Proposition 2
References
Strong I/O Lower Bounds for Binomial and FFT Computation Graphs
Introduction
Background
Boundary Flow Technique for Deriving Lower Bounds
Basic Technique
Constrained VIP for Deriving Better Bounds
Memory Traffic Complexity
Binomial Computation Graph
FFT Computation Graph
Conclusion
References
Spin Systems on Graphs with Complex Edge Functions and Specified Degree Regularities
Introduction
Overview of Techniques and Results
Background and Notation
Interpolation Technique
Classification of Complex Signatures
References
Quantum Algorithm for the Boolean Hidden Shift Problem
Introduction
Preliminaries
Our Algorithm
Classical Complexity of Random Instances of BHSP
Discussion and Open Problems
References
A Kolmogorov Complexity Proof of the Lov´asz Local Lemma for Satisfiability
Introduction
The Lovász Local Lemma
Preliminaries
The Kolmogorov Argument
The LOG
Witness Forests
Putting Things Together
An Improvement
Some Remarks on Unsatisfiable k-CNF Formulas
References
Proper n-Cell Polycubes in n −3 Dimensions
Introduction
Overview of the Method
Distinguished Structures
Polycubes with a Tree Structure
Polycubes with One Cycle
Polycubes with Two Cycles
Epilogue
Conclusion
References
Largest Area Convex Hull of Axis-Aligned Squares Based on Imprecise Data
Introduction
Previous Work
O(n^7) Algorithm for Problem 1
O(n^5) Algorithm for Problem 2
O(n^5) Algorithm for Problem 1
O(n^3) Algorithm for Problem 2
Conclusions
References
Improved Algorithms for the Point-Set Embeddability Problem for Plane 3-Trees
Introduction
Preliminaries
Algorithms
Analysis
For Points Not in General Positions
Conclusion
References
Optimal Strategies for the One-Round Discrete Voronoi Game on a Line
Introduction
Optimal Strategy of P2
Optimal Strategy of P1
An Important Lemma
A Characterization of the Optimal Placement by P1
Improving the Algorithm for the Optimal Strategy of P1
m=2
Extending to the General Case
Conclusions
References
Computing the Girth of a Planar Graph in Linear Time
Introduction
Preliminaries
Expanded Version, Density, and Contracted Graph
Outerplane Radius and Degree Reduction
Proving the Theorem by the Main Lemma
Dissection Tree, Nonleaf Problem, and Leaf Problem
Task 1: Computing a Dissection Tree
Task 2: Solving the Nonleaf Problems
A Reduction to the Border Problems for the Special Leaf Vertices
Solving the Border Problems for the Special Leaf Vertices
Task 3: Solving the Leaf Problem
Concluding Remarks
References
Diagonalization Strikes Back: Some Recent Lower Bounds in Complexity Theory
References
Unions of Disjoint NP-Complete Sets
Introduction
Preliminaries
Main Theorems
Length-Increasing Reductions
Discussion
References
ReachFewL = ReachUL
Introduction
Definitions and Necessary Results
ReachUL as an Oracle
Converting Graphs with a Few Paths to Distance Isolated Graphs
Converting Distance Isolated Graphs to Unambiguous Graphs
ReachFewL= ReachUL
Discussion
References
(1 + ε)-Competitive Algorithm for Online OVSF Code Assignment with Resource Augmentation
Introduction
Preliminaries
2-Competitive Algorithm with 2lg*h Trees
Preparation of Subtrees
Labeling Subtrees
Algorithm ALG_1
(1+1/α)-Competitive Algorithm with (1 + [α]) lg∗ h Trees
Preparation of Subtrees
Labeling Subtrees
Algorithm ALG_α
References
Scheduling Jobs on Heterogeneous Platforms
Introduction
An AFPTAS for SPP
Relaxed Schedule
Rounding the Fractional Solution
Strip Packing Subroutine
Analysis
Analyzing the Output
Running Time of the Algorithm
Malleable Jobs
Release Times
References
Self-assembling Rulers for Approximating Generalized Sierpinski Carpets
Introduction
Preliminaries
Embedded Fractals
Approximating Generalized Sierpinski Carpets
Rulers and Readers
Putting It all Together
Conclusion
References
Approximately Uniform Online Checkpointing
Introduction
Model
Checkpointing Algorithms
Powers-of-Two Algorithm
Golden Ratio Approach
Golden Ratio Algorithms
Optimal Algorithm for k=2
Lower Bound for Competitive Ratio
Conclusions and Future Work
References
Bandwidth of Convex Bipartite Graphs and Related Graphs
Introduction
Bandwidth of Convex Bipartite Graphs
NP-Completeness Result
Approximation Algorithms for Convex Bipartite Graphs
Bandwidth of 2-Directional Orthogonal Ray Graphs
Bandwidth of Biconvex Trees
References
Algorithms for Partition of Some Class of Graphs under Compaction
Introduction
Some Requirements for Compaction
Compaction to Reflexive and Irreflexive Paths
Compaction to Reflexive and Irreflexive Graphs through a Walk
An Algorithm for the Partition Problem COMP-H for Some Class of Input Graphs when H Is Reflexive
An Algorithm for the Partition Problem COMP-H for Some Class of Input Graphs when H Is Irreflexive
Structure of Graphs for Compaction to Cycles
References
A Generic Approach to Decomposition Algorithms, with an Application to Digraph Decomposition
Introduction
Efficient Tree Representations of Set Families
A Module-Split Digraph Decomposition
Splitmodules of Digraphs
A Tree Representation for Splitmodules
Conclusion
References
Matching and P_2-Packing: Weighted Versions
Introduction
Constrained P_2-Packings on Bipartite Graphs
Improved Algorithms for the P_2-Packing Problem
Conclusion
References
On Totally Unimodularity of Edge-Edge Adjacency Matrices
Introduction
Preliminaries
A Necessary and Sufficient Condition on Totally Unimodularity of EE Matrices
Necessary Condition
Sufficient Condition
Conclusion
References
The Topology Aware File Distribution Problem
Introduction
Related Work
The Router Topology Aware (RTA) Network Model
An O(logn)-Approximation
Max-Min Edge Workstation Spanning Tree, MMWST
Tree Splitting, TREESPLIT
NP-Completeness
Conclusion
References
Exploiting the Robustness on Power-Law Networks
Introduction
Models, Measurement and Threat Taxonomy
Power-Law Random Graph Model (PLRG)
Vulnerability Measurements
Threat Taxonomy
Preliminaries
Uniform Random Failures
Preferential Attacks
Interactive Attacks ( pi = 1- 1/i^β')
Expected Attacks ( pi = c i/e^αζ(β−1) )
Degree-Centrality Attacks
References
Competitive Algorithms for Online Pricing
Introduction
h Is Known in Advance
Lower Bound
Online Algorithm
h Is Not Known in Advance
References
Making Abstraction-Refinement EÆcient in Model Checking
Introduction
Related Work
Abstraction Function
Refinement
Why Refining?
Spurious Counterexamples
Refining Algorithm
Abstract Model Checking Framework
Conclusion
References
An Integer Programming Approach for the Rural Postman Problem with Time Dependent Travel Times
Introduction
Problem Formulation
Results on RPPTDT Polyhedron
Computational Results
Application in Scheduling with Time Dependent Processing Times
Conclusions
References
Property Testing for Cyclic Groups and Beyond
Introduction
Definitions
A Lower Bound for Testing Cyclic Groups
A Lower Bound for Testing the Number of Generators in a Group
Testing If the Input Is Cyclic When |Γ| Is Known
References
Canonizing Hypergraphs under Abelian Group Action
Introduction
Complexity Classes
Preliminary Observations
Canonizing Hypergraphs with Color Classes of Size 2
Canonizing Hypergraphs under Abelian Groups
References
Linear Time Algorithms for the Basis of Abelian Groups
Introduction
Preliminaries
Our Results
Auxiliary Results
The Extended Discrete Logarithm Problem
The BASIS Algorithm
The ORDER-FINDING Algorithm
Computing the Order of an Element in O((logN)2loglogN)-Time
Presentation of Our Algorithms
First Proof of O(N)-Time Algorithm
Second Proof of O(N)-Time Algorithm
Proof of Corollary 1
Conclusion
References
Characterizations of Locally Testable Linear andAffine-Invariant Families
Introduction
Theorems and Methods
Constraints on Affine/Linear-Invariant Families
Characterizations for Linear-Invariant Families
Generalization to Extension Fields
References
A New Conditionally Anonymous Ring Signature
Introduction
Preliminaries
Model of Conditionally Anonymous Ring Signature
Syntax
Oracles
Security Model
Construction
Performance
Security
Conclusion
References
On the Right-Seed Array of a String
Definitions and Problems
Properties
The Algorithms
A Brief Description of the Partitioning Algorithm
Computing the Minimal and the Maximal Right-Seed Array
Further Works
References
Compressed Directed Acyclic Word Graph with Application in Local Alignment
Introduction
Basic Concepts and Definitions
Suffix Array and Suffix Tree
Advanced Data-Structures for Suffix Array and Suffix Tree
Directed Acyclic Word Graph
Simulating DAWG
Get-Source Operation
End-Set Operations
Child Operation
Parent Operations
Application of DAWG in Local Alignment
Definitions of Global, Local, and Meaningful Alignments
Local Alignment Using DAWG
References
Unavoidable Regularities in Long Words with Bounded Number of Symbol Occurrences
Introduction
Basics on Alphabets and Words
The Main Theorem
Further Considerations on Permutations
Connections to Multicollision Attacks on Generalized Iterated Hash Functions
Generalization in Practice
Conclusion
References
Summing Symbols in Mutual Recurrences
Problem Statement
Related Work
Systems of Mutual Recurrences
Homogeneous Case
Inhomogeneous Case
The Summation Procedure
Analysis
Conclusion
References
Flipping Triangles and Rectangles
Introduction
Flipping Triangles
Chromatic Number
Clique-Partition Number
Independence Number
Imperfection Ratio and Fractional Chromatic Number
Flipping Rectangles
Rolling Blocks
Open Question
References
Unconstained and Constrained Fault-Tolerant Resource Allocation
Introduction
Unconstrained FTRA
The Algorithms
Analysis: Dual Fitting and Inverse Dual Fitting
Capacitated UFTRA
Constrained FTRA
References
Finding Paths with Minimum Shared Edges
Introduction
NP-Hardness Proof
Approximation Algorithm
Inapproximability Result
Heuristic Improvements
Successive Cost Update
Shortest Path Bound
Experimental Results
Conclusions
References
Combinatorial Group Testing for Corruption Localizing Hashing
Introduction
Model and Preliminaries
Using Localizing Codes for Corruption Localization
A First Localizing Code
A Localizing Code Achieving Constant Localization Factor
References
Task Ordering and Memory Management Problem for Degree of Parallelism Estimation
Introduction
Related Work
Problem Formulation and Complexity
Memory Management for a Fixed Sequence of Tasks
Task Ordering and Memory Management
Branching Rule
Lower Bounds
Dominance Relation
Selection Rules
Computational Results
Conclusion
References
Computing Majority with Triple Queries
Introduction
Model of Computation
Notation
Related Work
Outline and Results of the Paper
Existence of Solutions
Y/N Model
The Upper Bound
The Lower Bound
Pairing Model
A General Lower Bound
Determining q_3^p (n)
Conclusion
References
A New Variation of Hat Guessing Games
Introduction
Preliminaries
(d1,d2)-Regular Partition of Q_n
The Maximum Winning Probability P_n,k
Conclusion
References
Oblivious Transfer and n-Variate Linear Function Evaluation
Introduction
Preliminaries
Statistically Secure Reduction from (n 1)-OT to C-OLFE_n
C-OLFEn and Reversing (n 1)-OT
Conclusion
References
Optimal Online Algorithms on Two Hierarchical Machines with Resource Augmentation
Introduction
Optimal Algorithm for 01
Conclusions
References
Author Index

Citation preview

Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max Planck Institute for Informatics, Saarbruecken, Germany

6842

Bin Fu Ding-Zhu Du (Eds.)

Computing and Combinatorics 17th Annual International Conference, COCOON 2011 Dallas, TX, USA, August 14-16, 2011 Proceedings

13

Volume Editors Bin Fu University of Texas-Pan American Department of Computer Science Edinburg, TX 78539, USA E-mail: [email protected] Ding-Zhu Du University of Texas at Dallas Department of Computer Science Richardson, TX 75080, USA E-mail: [email protected]

ISSN 0302-9743 e-ISSN 1611-3349 ISBN 978-3-642-22684-7 e-ISBN 978-3-642-22685-4 DOI 10.1007/978-3-642-22685-4 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011932219 CR Subject Classification (1998): F.2, C.2, G.2, F.1, E.1, I.3.5 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues

© Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The Annual International Computing and Combinatorics Conference is a forum for researchers working in the areas of algorithms, theory of computation, computational complexity, and combinatorics related to computing. The papers in this volume were presented at the 17th Annual International Computing and Combinatorics Conference (COCOON 2011), held in the city of Dallas, Texas, USA, during August 14-16, 2011. Previous meetings of this conference were held in Singapore (2002), Big Sky (2003), Jeju Island (2004), Kunming (2005), Taipei (2006), Alberta (2007), Dalian (2008), New York (2009), and NhaTrang (2010). A total of 136 papers were submitted, of which 54 were accepted for presentation at the conference. We received papers from Austrialia, Brazil, Canada, China, Denmark, Finland, France, Germany, Greece, Hong Kong, India, Iran, Israel, Italy, Japan, Korea, Mexico, Norway, Poland, Singapore, Spain, Taiwan, UK, and USA. The papers were evaluated by an international Program Committee consisting of Hee-Kap Ahn, Tatsuya Akutsu, Eric Allender, Zhixiang Chen, Zhi-zhong Chen, Qi Cheng, Ding-Zhu Du, Bin Fu, Xiaofeng Gao, Wen-Lian Hsu, Kazuo Iwama, Iyad Kanj, Neeraj Kayal, Ming-Yang Kao, Donghyun Kim, D. T. Lee, Angsheng Li, Pinyan Lu, Jack Lutz, Mitsunori Ogihara, Hi-rotaka Ono, Desh Ranjan, David Sankoff, Kavitha Telikepalli, Carola Wenk, Boting Yang, Louxin Zhang, and Shengyu Zhang. Each paper was evaluated by at least three Program Committee members, assisted in some cases by external referees. The selection was based on the papers’ originality, quality, and relevance to topics of the COCOON 2011. It is expected that most of the accepted papers will appear in a more polished form in scientific journals. In addition to the selected papers, the conference also included one invited presentation by Ryan Williams. The Program Committee selected “Unions of Disjoint NP-complete Sets” by Christian Glaßer, John Hitchcocky, A. Pavan, and Stephen Travers for the Best Paper Award. We thank all the people who made this meeting possible: the authors for submitting papers, the Program Committee members and external referees for their excellent work, and the invited speaker. Finally, we thank the colleagues at the University of Texas at Dallas for their local arrangements and assistance. August 2011

Bin Fu Ding-Zhu Du

Organization

Executive Committee Conference TPC Chairs

Local Arrangements Chair

Bin Fu (University of Texas - Pan American, USA) Ding-Zhu Du (University of Texas at Dallas, USA) Weili Wu (University of Texas at Dallas, USA)

Program Committee Hee-Kap Ahn Tatsuya Akutsu Eric Allender Zhixiang Chen Zhi-zhong Chen Qi Cheng Ding-zhu Du Bin Fu Xiaofeng Gao Wen-Lian Hsu Kazuo Iwama Iyad Kanj Neeraj Kayal Ming-Yang Kao Donghyun Kim D.T. Lee Angsheng Li Pinyan Lu Jack Lutz Mitsunori Ogihara Hirotaka Ono Desh Ranjan David Sankoff Kavitha Telikepalli Carola Wenk Boting Yang Louxin Zhang Shengyu Zhang

Pohang University of Science and Technology, Korea Kyoto University, Japan Rugters University, USA University of Texas-Pan American, USA Tokyo Denki University, Japan Univeristy of Oklahoma, USA University of Texas at Dallas, USA, Co-chair University of Texas-Pan American, USA, Co-chair Georgia Gwinnett College, USA Academia Sinica, Taiwan Kyoto University, Japan DePaul University, USA Microsoft, USA Northwestern University, USA North Carolina Central University, USA Academia Sinica, Taiwan Chinese Academia of Science, China Beijing Microsoft Lab, China Iowa State Univeristy, USA University of Miami, USA Kyushu University, Japan Old Dominion University, USA University of Ottawa, Canada Tata Institute of Fundamental Research, India Univeristy of Texas at San Antonio, USA University of Regina, Canada University of Singapore, Singapore Chinese University of Hong Kong, Hong Kong

VIII

Organization

Referees Ferdinando Cicalese Paolo D’Arco Yuan-Shin Lee

Zaixin Lu Gaolin Milledge Seth Pettie

Salvatore La Torre Lidong Wu Jiaofei Zhong

Invited Speaker Ryan Williams

IBM T.J. Walson Research Center, USA

Table of Contents

Derandomizing HSSW Algorithm for 3-SAT . . . . . . . . . . . . . . . . . . . . . . . . . Kazuhisa Makino, Suguru Tamaki, and Masaki Yamamoto

1

Dominating Set Counting in Graph Classes . . . . . . . . . . . . . . . . . . . . . . . . . . Shuji Kijima, Yoshio Okamoto, and Takeaki Uno

13

The Density Maximization Problem in Graphs . . . . . . . . . . . . . . . . . . . . . . . Mong-Jen Kao, Bastian Katz, Marcus Krug, D.T. Lee, Ignaz Rutter, and Dorothea Wagner

25

FlipCut Supertrees: Towards Matrix Representation Accuracy in Polynomial Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Malte Brinkmeyer, Thasso Griebel, and Sebastian B¨ ocker

37

Tight Bounds on Local Search to Approximate the Maximum Satisfiability Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daming Zhu, Shaohan Ma, and Pingping Zhang

49

Parameterized Complexity in Multiple-Interval Graphs: Partition, Separation, Irredundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minghui Jiang and Yong Zhang

62

Exact Parameterized Multilinear Monomial Counting via k-Layer Subset Convolution and k-Disjoint Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dongxiao Yu, Yuexuan Wang, Qiang-Sheng Hua, and Francis C.M. Lau On the Rainbow Connectivity of Graphs: Complexity and FPT Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kei Uchizawa, Takanori Aoki, Takehiro Ito, Akira Suzuki, and Xiao Zhou

74

86

On Parameterized Independent Feedback Vertex Set . . . . . . . . . . . . . . . . . . Neeldhara Misra, Geevarghese Philip, Venkatesh Raman, and Saket Saurabh

98

Cograph Editing: Complexity and Parameterized Algorithms . . . . . . . . . . Yunlong Liu, Jianxin Wang, Jiong Guo, and Jianer Chen

110

Approximation Complexity of Complex-Weighted Degree-Two Counting Constraint Satisfaction Problems (Extended Abstract) . . . . . . . Tomoyuki Yamakami

122

X

Table of Contents

Strong I/O Lower Bounds for Binomial and FFT Computation Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Desh Ranjan, John Savage, and Mohammad Zubair

134

Spin Systems on Graphs with Complex Edge Functions and Specified Degree Regularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jin-Yi Cai and Michael Kowalczyk

146

Quantum Algorithm for the Boolean Hidden Shift Problem . . . . . . . . . . . . Dmitry Gavinsky, Martin Roetteler, and J´er´emie Roland A Kolmogorov Complexity Proof of the Lov´asz Local Lemma for Satisfiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jochen Messner and Thomas Thierauf Proper n-Cell Polycubes in n – 3 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . Andrei Asinowski, Gill Barequet, Ronnie Barequet, and G¨ unter Rote

158

168 180

Largest Area Convex Hull of Axis-Aligned Squares Based on Imprecise Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ovidiu Daescu, Wenqi Ju, Jun Luo, and Binhai Zhu

192

Improved Algorithms for the Point-Set Embeddability Problem for Plane 3-Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tanaeem M. Moosa and M. Sohel Rahman

204

Optimal Strategies for the One-Round Discrete Voronoi Game on a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aritra Banik, Bhaswar B. Bhattacharya, and Sandip Das

213

Computing the Girth of a Planar Graph in Linear Time . . . . . . . . . . . . . . Hsien-Chih Chang and Hsueh-I. Lu Diagonalization Strikes Back: Some Recent Lower Bounds in Complexity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ryan Williams

225

237

Unions of Disjoint NP-Complete Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christian Glaßer, John M. Hitchcock, A. Pavan, and Stephen Travers

240

ReachFewL = ReachUL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brady Garvin, Derrick Stolee, Raghunath Tewari, and N.V. Vinodchandran

252

(1 + ε)-Competitive Algorithm for Online OVSF Code Assignment with Resource Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuichi Asahiro, Kenta Kanmera, and Eiji Miyano Scheduling Jobs on Heterogeneous Platforms . . . . . . . . . . . . . . . . . . . . . . . . Marin Bougeret, Pierre Francois Dutot, Klaus Jansen, Christina Robenek, and Denis Trystram

259 271

Table of Contents

Self-assembling Rulers for Approximating Generalized Sierpinski Carpets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steven M. Kautz and Brad Shutters

XI

284

Approximately Uniform Online Checkpointing . . . . . . . . . . . . . . . . . . . . . . . Lauri Ahlroth, Olli Pottonen, and Andr´e Schumacher

297

Bandwidth of Convex Bipartite Graphs and Related Graphs . . . . . . . . . . . Anish Man Singh Shrestha, Satoshi Tayu, and Shuichi Ueno

307

Algorithms for Partition of Some Class of Graphs under Compaction . . . Narayan Vikas

319

A Generic Approach to Decomposition Algorithms, with an Application to Digraph Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binh-Minh Bui-Xuan, Pinar Heggernes, Daniel Meister, and Andrzej Proskurowski

331

Matching and P2 -Packing: Weighted Versions . . . . . . . . . . . . . . . . . . . . . . . . Qilong Feng, Jianxin Wang, and Jianer Chen

343

On Totally Unimodularity of Edge-Edge Adjacency Matrices . . . . . . . . . . Yusuke Matsumoto, Naoyuki Kamiyama, and Keiko Imai

354

The Topology Aware File Distribution Problem . . . . . . . . . . . . . . . . . . . . . . Shawn T. O’Neil, Amitabh Chaudhary, Danny Z. Chen, and Haitao Wang

366

Exploiting the Robustness on Power-Law Networks . . . . . . . . . . . . . . . . . . . Yilin Shen, Nam P. Nguyen, and My T. Thai

379

Competitive Algorithms for Online Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . Yong Zhang, Francis Y.L. Chin, and Hing-Fung Ting

391

Making Abstraction-Refinement Efficient in Model Checking . . . . . . . . . . Cong Tian and Zhenhua Duan

402

An Integer Programming Approach for the Rural Postman Problem with Time Dependent Travel Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guozhen Tan and Jinghao Sun

414

Property Testing for Cyclic Groups and Beyond . . . . . . . . . . . . . . . . . . . . . Fran¸cois Le Gall and Yuichi Yoshida

432

Canonizing Hypergraphs under Abelian Group Action . . . . . . . . . . . . . . . . V. Arvind and Johannes K¨ obler

444

XII

Table of Contents

Linear Time Algorithms for the Basis of Abelian Groups . . . . . . . . . . . . . . Gregory Karagiorgos and Dimitrios Poulakis Characterizations of Locally Testable Linear- and Affine-Invariant Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Angsheng Li and Yicheng Pan

456

467

A New Conditionally Anonymous Ring Signature . . . . . . . . . . . . . . . . . . . . Shengke Zeng, Shaoquan Jiang, and Zhiguang Qin

479

On the Right-Seed Array of a String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michalis Christou, Maxime Crochemore, Ondrej Guth, Costas S. Iliopoulos, and Solon P. Pissis

492

Compressed Directed Acyclic Word Graph with Application in Local Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Do Huy Hoang and Sung Wing Kin

503

Unavoidable Regularities in Long Words with Bounded Number of Symbol Occurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Juha Kortelainen, Tuomas Kortelainen, and Ari Vesanen

519

Summing Symbols in Mutual Recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . Berkeley R. Churchill and Edmund A. Lamagna

531

Flipping Triangles and Rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minghui Jiang

543

Unconstrained and Constrained Fault-Tolerant Resource Allocation . . . . Kewen Liao and Hong Shen

555

Finding Paths with Minimum Shared Edges . . . . . . . . . . . . . . . . . . . . . . . . . Masoud T. Omran, J¨ org-R¨ udiger Sack, and Hamid Zarrabi-Zadeh

567

Combinatorial Group Testing for Corruption Localizing Hashing . . . . . . . Annalisa De Bonis and Giovanni Di Crescenzo

579

Task Ordering and Memory Management Problem for Degree of Parallelism Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sergiu Carpov, Jacques Carlier, Dritan Nace, and Renaud Sirdey

592

Computing Majority with Triple Queries . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gianluca De Marco, Evangelos Kranakis, and G´ abor Wiener

604

A New Variation of Hat Guessing Games . . . . . . . . . . . . . . . . . . . . . . . . . . . Tengyu Ma, Xiaoming Sun, and Huacheng Yu

616

Table of Contents

Oblivious Transfer and n-Variate Linear Function Evaluation . . . . . . . . . . Yeow Meng Chee, Huaxiong Wang, and Liang Feng Zhang

XIII

627

Optimal Online Algorithms on Two Hierarchical Machines with Resource Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yiwei Jiang, An Zhang, and Jueliang Hu

638

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

649

Derandomizing HSSW Algorithm for 3-SAT Kazuhisa Makino1 , Suguru Tamaki2 , and Masaki Yamamoto3 1

Graduate School of Information Science and Technology, University of Tokyo [email protected] 2 Graduate School of Informatics, Kyoto University [email protected] 3 Dept. of Informatics, Kwansei-Gakuin University [email protected]

Abstract. We present a (full) derandomization of HSSW algorithm for 3-SAT, proposed by Hofmeister, Sch¨ oning, Schuler, and Watanabe in n  )-time deterministic algo[STACS’02]. Thereby, we obtain an O(1.3303 rithm for 3-SAT, which is currently fastest.

1

Introduction

The satisfiability problem (SAT) is one of the most fundamental NP-hard problems. Questing for faster (exponential-time) exact algorithms is one of the main research directions on SAT. Initiated by Monien and Speckenmeyer [12], a number of algorithms for exactly solving SAT have been proposed, and many important techniques to analyze those algorithms have been developed [6]. See also [5, 13, 15, 16, 20], for example. The most well-studied restriction of the satisfiability problem is 3-SAT [2, 3, 7–10, 18, 19, 21], i.e., the CNF satisfiability problem with clauses of length at most three. The currently best known time n  ) achieved by randomized algorithms [7] complexities for 3-SAT are O(1.3211 n  and O(1.3334 ) derived by deterministic algorithms [13], where n denotes the  in the number of Boolean variables in the formula. (We use the notation O  ordinary way, that is, O(f (n)) means O(poly(n)f (n)).) As we can see, there is a noticeable gap between the current randomized and deterministic time bounds for 3-SAT. This raises a natural question: Can we close the gap completely? One promising way to attack the above question is derandomization. Roughly speaking, the task of derandomization is to construct an algorithm which deterministically and efficiently simulates the original randomized algorithm. There are a lot of strong derandomization results, e.g. [1, 4, 14, 17] to name a few, and one excellent example in the area of satisfiability is the derandomization of Sch¨ oning’s algorithm for k-SAT. In [20], Sch¨ oning proposed a simple randomized local search algorithm for  k-SAT, and showed that it runs in expected time O((2 − 2/k)n ), which is n  O(1.3334 ) when k = 3. Later it was derandomized by Dantsin et al. [5]. They proposed a k-SAT algorithm that deterministically simulates Sch¨ oning’s algon   ) when k = 3. Sch¨ oning’s alrithm in time O((2 − 2/(k + 1))n ), which is O(1.5 gorithm makes use of randomness in the following two parts: (i) choosing initial B. Fu and D.-Z. Du (Eds.): COCOON 2011, LNCS 6842, pp. 1–12, 2011. c Springer-Verlag Berlin Heidelberg 2011 

2

K. Makino, S. Tamaki, and M. Yamamoto

assignments for local search uniformly at random, and (ii) performing random walks as the local search. Dantsin et al. [5] derandomized it (i) by constructing a set of Hamming balls (so-called covering codes), which efficiently covers the entire search space {0, 1}n, and (ii) by replacing each random walk by backtracking search. Here (i) is “perfectly” derandomized in some sense, however, the derandomization of (ii) loses some efficiency. For 3-SAT, the efficiency in derandomizing part (ii) was gradually improved by a sequence of works [3, 5, 11, 21]. Finally, and very recently, Moser and Scheder [13] showed a full derandomization of Sch¨ oning’s algorithm, that is, they proposed a deterministic algorithm for k SAT that runs in time O((2−2/k +)n) for any  > 0. The running time matches n  ) time that of Sch¨ oning’s algorithm, and we now have a deterministic O(1.3334 algorithm for 3-SAT. Our Contribution We investigate the possibility of derandomizing faster randomized algorithms for 3-SAT. In [8], Hofmeister, Sch¨oning, Schuler and Watanabe improved Sch¨ oning’s algorithm for the 3-SAT case, that is, they proposed a randomized algorithm for n  ). Their improvement is based on 3-SAT that runs in expected time O(1.3303 a sophisticated way of randomly choosing initial assignments rather than just choosing the ones uniformly at random. In this paper, we present a full derandomization of their algorithm, that immediately implies the following result: n  Theorem 1. Problem 3-SAT is deterministically solvable in time O(1.3303 ). As long as the authors know, it is the currently fastest deterministic algorithm for 3-SAT. Our result seems to be a necessary step towards a full derandomization of the currently best known randomized algorithm, since it is based on the combination of two algorithms [9] and [7], which are respectively a modification of Hofmeister et al.’s algorithm [8] and an extension of Paturi et al.’s algorithm [15]. To prove the above result, we develop a new way of explicitly constructing covering codes with the properties which corresponds to the distribution used to generate initial assignments in Hofmeister et al.’s algorithm. More precisely, we respectively denote by SCH and HSSW the randomized algorithms by Sch¨ oning [20], and by Hofmeister, Sch¨ oning, Schuler, and Watanabe [8]. Algorithm HSSW is obtained by modifying SCH, where one of the main differences between SCH and HSSW is to choose initial assignments for random walks as the local search: HSSW starts the random walk at an assignment choˆ for some m ˆ ≤ n/3, while SCH starts it at an sen randomly from ({0, 1}3 \ 03 )m assignment chosen uniformly from the whole space {0, 1}n. We derandomized this random selection of initial assignments for HSSW in the similar way to SCH [5], i.e., by constructing a covering code (i.e., a set of ˆ ). However, due to the balls that covers the whole search space ({0, 1}3 \ 03 )m 3 3 m ˆ n difference of ({0, 1} \ 0 ) and {0, 1} , we cannot directly apply a uniform ˆ covering code developed in [5]. To efficiently cover the space ({0, 1}3 \ 03 )m , we

Derandomizing HSSW Algorithm for 3-SAT

3

introduced a generalized covering code, an []-covering code, which is a sequence of codes C(0), C(1), . . . , C() such that (i) C(i) is a set of balls of radius i, and (ii)  3 3 m ˆ i=0 C(i) covers ({0, 1} \ 0 ) . We remark that the generalized covering code has non-uniform covering radius while an ordinary covering code has uniform radius. We first show the existence of small []-covering code (C(0), C(1), . . . , C()), and then similarly to [5], by using an approximation algorithm for the set cover problem, we show a deterministic construction of an []-covering code ˜ ˜ ˜ ˜ C(0), C(1), . . . , C() such that |C(i)| ≈ |C(i)|. We remark that our technique of constructing certain types of covering codes has a potential application, for example, it can be applied to the further extensions [2, 18] of HSSW.

2

Preliminaries

In this section, we briefly review HSSW algorithm for 3-SAT proposed in [8]. In what follows, we focus on 3-CNF formulas. Let ϕ be a 3-CNF formula over X = {x1 , . . . , xn }. We alternatively regard ϕ as the set of clauses of ϕ. Thus, the size of ϕ, which is the number of clauses of ϕ, is denoted by |ϕ|. For any sub-formula ϕ ⊂ ϕ (resp., any clause C ∈ ϕ), we denote by X(ϕ ) (resp., X(C)) the set of variables of ϕ (resp., C). A clause set ϕ ⊂ ϕ is independent if X(C) ∩ X(C  ) = ∅ for any pair of clauses C, C  ∈ ϕ . An independent clause set ϕ is maximal if for any clause C ∈ (ϕ \ ϕ ) there exists a clause C  ∈ ϕ such that X(C) ∩ X(C  ) = ∅. For any partial assignment t to X(ϕ), we denote by ϕ|t a sub-formula obtained from ϕ by fixing variables according to t. Given a 3-CNF formula ϕ, algorithm HSSW starts with arbitrarily finding a maximal independent clause set of ϕ. Fact 1. Let ϕ be a 3-CNF formula. Let ϕ ⊂ ϕ be a maximal independent clause set of ϕ. Then, for any assignment t to X(ϕ ), the formula ϕ|t is a 2-CNF formula. Before describing HSSW, we briefly review SCH algorithm for k-SAT proposed in [20]. Algorithm SCH is a randomized algorithm which repeats the following procedure exponentially (in n) many times: choose a random assignment t, and run a random walk starting at t as follows: for a current assignment t , if ϕ is satisfied by t , then output YES and halt. Otherwise, choose an arbitrary clause C unsatisfied by t , and then update t by flipping the assignment of a variable of C chosen uniformly at random. This random walk procedure denoted by SCH-RW(ϕ, t) is also exploited in HSSW. The success probability of SCH-RW(ϕ, t) for a satisfiable ϕ was analyzed in [20]: Let ϕ be a 3-CNF formula that is satisfiable. Let t0 be an arbitrary satisfying assignment of ϕ. Then, for any initial assignment t with Hamming distance d(t0 , t) = r, we have  r 1 1 . (1) · Pr{SCH-RW(ϕ, t) = YES} ≥ 2 poly(n)

4

K. Makino, S. Tamaki, and M. Yamamoto

Now, we are ready to present HSSW. Given a 3-CNF formula ϕ, HSSW first obtains a maximal independent clause set ϕ ⊂ ϕ. Note here that the formula ϕ|t for any assignment to X(ϕ ) is a 2-CNF, and hence we can check in polynomial time whether ϕ|t is satisfiable. From this observation, when ϕ is small, we can  |ϕ | ) significantly improve the whole running time, that is, it only requires O(7 time. On the other hand, when the size of ϕ is large, we repeatedly apply the random walk procedure SCH-RW. In this case, we can also reduce the running time by smartly choosing initial assignments from satisfiable assignments of ϕ : Recall that SCH uniformly chooses initial assignments from {0, 1}n, which utilizes no information on ϕ. Intuitively, HSSW uses initial assignments for SCH-RW that are closer to any satisfiable assignment. In fact we can prove that the larger the size of ϕ is, the higher the probability that the random walk starts at an assignment closer to a satisfying assignment is. Formally, algorithm HSSW is described in Fig. 1. The algorithm contains 5 parameters α, c, and triple (a1 , a2 , a3 ) with 3a1 +3a2 +a3 = 1. These parameters are set to minimize the whole expected running time. HSSW(ϕ)

// ϕ: a 3-CNF formula over X

Obtain a maximal independent clause set ϕ ⊂ ϕ If |ϕ | ≤ αn, then  for each t ∈ {0, 1}X(ϕ ) that satisfies ϕ Check the satisfiability of ϕ|t // ϕ|t : a 2-CNF formula If |ϕ | > αn, then c times do Run t = init-assign(X, ϕ ) Run SCH-RW(ϕ, t) // YES is output here if a solution is found Output NO init-assign(X, ϕ ) follows

// return an assignment t ∈ {0, 1}X defined as

for each C ∈ ϕ Assume C = xi ∨ xj ∨ xk Choose a random assignment t to x = (xi , xj , xk ) following the probability distribution: Pr{x = (1, 0, 0)} = Pr{x = (0, 1, 0)} = Pr{x = (0, 0, 1)} = a1 Pr{x = (1, 1, 0)} = Pr{x = (1, 0, 1)} = Pr{x = (0, 1, 1)} = a2 Pr{x = (1, 1, 1)} = a3 for each x ∈ X \ X(ϕ ) Choose a random assignment t to x ∈ {0, 1} Fig. 1. Algorithm HSSW

Derandomizing HSSW Algorithm for 3-SAT

5

Consider algorithm HSSW in Fig. 1 when |ϕ | > αn for some constant α > 0. (In what follows, we focus on this case since for the other case, it has no randomness.) Let HSSW-RW(ϕ, ϕ ) be the procedure that is repeated c times. Then, by using the lower bound (1), and setting parameters (a1 , a2 , a3 ) suitably (c.f., Lemma 1 below), we have: for any satisfiable 3-CNF formula ϕ,  n  |ϕ | 3 64 · . Pr {HSSW-RW(ϕ, ϕ ) = YES} ≥ t,SCH-RW 4 63 

(2)

n  ) is obtained by setting α to satisfy The whole expected running time O(1.3303 the following equation.

 n  αn −1 3 64 · = 7αn . 4 63 The values of parameters (a1 , a2 , a3 ) are determined according to the following lemma, which will be used by our derandomization. Lemma 1 (Hofmeister, Sch¨ oning, Schuler, and Watanabe [8]). Let ϕ be a 3-CNF formula that is satisfiable, and let ϕ ⊂ ϕ be a maximal independent clause set of ϕ. Let t be a random (partial) assignment obtained via init-assign(X, ϕ ) and restricted to X(ϕ ). Then, for any (partial) assignment  t0 ∈ {0, 1}X(ϕ ) that satisfies ϕ ,       d(t0 ,t) |ϕ | 1 3 . (3) = E t 2 7 There are two types of randomness that are used in HSSW: (1) the random assignment obtained via init-assign, and (2) the random walk of SCH-RW. Fortunately, the latter type of randomness can be (fully) removed by the recent result. (Compare it with the inequality (1).) Theorem 2 (Moser and Scheder [13]). Let ϕ be a 3-CNF formula that is satisfiable. Given an assignment t such that there exists a satisfying assignment t0 of ϕ such that d(t0 , t) = r for a non-negative integer r. Then, a satisfying assignment (not necessarily to be t0 ) can be found deterministically in time  O((2 + )r ) for any constant  > 0. In the next section, we show that the former type of randomness is also not necessary. It is shown by using covering codes, that is in the similar way to [5]. But, the covering code we make use of is different from ordinary ones. For any positive integer n, a code of length n is a subset of {0, 1}n , where each element of a code is called a codeword. A code C ⊂ {0, 1}n is called an r-covering code if for every x ∈ {0, 1}n, there exists a codeword y ∈ C such that d(x, y) ≤ r. This is the definition of an ordinary covering code. We define a generalization of covering codes in the following way:

6

K. Makino, S. Tamaki, and M. Yamamoto

Definition 1. Let  be a non-negative integer. A sequence C(0), C(1), . . . , C() of codes is a {0, 1, . . . , }-covering code, or simply an []-covering code, if for every x ∈ {0, 1}n, there exists a codeword y ∈ C(r) for some r : 0 ≤ r ≤  such that d(x, y) ≤ r. For a set S of non-negative integers, a sequence (C(i) : i ∈ S) of codes is called an S-covering code. For ordinary covering codes, it is easy to show the existence of a “good” rcovering code. Moreover, it is known that we can deterministically construct such an r-covering code. Lemma 2 (Dantsin et al. [5]). Let d ≥ 2 be a constant that divides n ≥ 1, and let 0 < ρ < 1/2. Then, there is a polynomial qd (n) such that a covering code of length n, radius at most ρn, and size at most qd (n)2(1−h(ρ))n , can be deterministically constructed in time qd (n)(23n/d +2(1−h(ρ))n ). The function h(x) is the binary entropy function, that is, h(x) = −x log2 x − (1 − x) log2 (1 − x) for 0 < x < 1.

3

A Derandomization of HSSW

In this section, we prove Theorem 1 by derandomizing HSSW. We do that in the similar way to [5]. Let ϕ be a 3-CNF formula, and ϕ be a maximal independent ˆ We suppose m ˆ = Ω(n) since we focus on the case of clause set of ϕ. Let |ϕ | = m. |ϕ | > αn. As is explained in the Introduction, we will use a generalized covering code: an []-covering code. First, we show that there exists an []-covering code ˆ where each of its codes is of small size. for ({0, 1}3 \ 03 )m ˆ Lemma 3. For ({0, 1}3 \ 03 )m , there exists an []-covering code C(0), C(1), . . . , ˆ C(), where  is the maximum integer such that (3/7)m < (1/2)−2 , and |C(i)| = 2 m ˆ i O(m ˆ (7/3) /2 ).

Proof. We show the existence of such an []-covering code by a probabilistic argument, as is the case of the existence of an ordinary covering code for {0, 1}n. However, the probabilistic construction of an []-covering code is different from the simple one of an ordinary covering code in terms of, (1) non-uniform covering radius, and (2) non-uniform choice of codewords. For obtaining the desired covering code, we make use of the probability distribution calculated in [8], that is, the equation (3) of Lemma 1. The probabilistic construction is as follows: Let  be the integer defined above. For each ˆ be a random code obtained by choosi : 0 ≤ i ≤ , let C(i) ⊂ ({0, 1}3 \ 03 )m 3 3 m ˆ ing y ∈ ({0, 1} \ 0 ) according to the distribution defined by the function ˆ /2i init-assign (in Fig. 1), and by repeating it independently s(i) = 8m ˆ 2 (7/3)m times. Note here that |C(i)| ≤ s(i). We will show that C(0), C(1), . . . , C() is an []-covering code with high probˆ arbitrarily. Note here that  ≤ 2m ˆ and (1/2)−1 ≤ ability. Fix x ∈ ({0, 1}3 \ 03 )m m ˆ (3/7) . Then, 3m ˆ  i=0

(1/2)i Pr{d(x, y) = i} y

Derandomizing HSSW Algorithm for 3-SAT

=

 

(1/2)i Pr{d(x, y) = i} + y

i=0

≤

 

≤

(1/2)i Pr{d(x, y) = i} y

i=+1

(1/2)i Pr{d(x, y) = i} + (1/2) y

i=0  

3m ˆ 

7

ˆ (1/2)i Pr{d(x, y) = i} + (3/7)m /2. y

i=0

Recall from the equation (3) of Lemma 1 that,     m 3m ˆ d(x,y) ˆ  3 1 i E (1/2) Pr{d(x, y) = i} = . = y y 2 7 i=0 From these two, we have   ˆ (1/2)i Pr{d(x, y) = i} ≥ (3/7)m /2. i=0

y

From this, we see there exists an r : 0 ≤ r ≤  such that ˆ r−1 2 /( + 1). Pr{d(x, y) = r} ≥ (3/7)m y

(4)

ˆ Note that this value of r depends on x. Thus, for each x ∈ ({0, 1}3 \ 03 )m , if we define

def rx = arg max (1/2)i Pr{d(x, y) = i} , i:0≤i≤

we see that r = rx satisfies the above inequality (4) 1 . Let B(z, i) be the set ˆ such that d(z, w) ≤ i. Then, from the lower bound (4), the of w ∈ {0, 1}3m probability that x is not covered with any C(i) is ⎫ ⎫ ⎧ ⎧  ⎬ ⎬ ⎨ ⎨    Pr x ∈ B(z, i) ≤ Pr B(z, rx ) x ∈ C ⎩ ⎭ C(rx ) ⎩ ⎭ i=0 z∈C(i)

z∈C(rx )

= Pr {∀y ∈ C(rx )[d(x, y) > rx ]} C(rx )

s(rx )  = Pr {[d(x, y) > rx ]} y

 s(rx ) = 1 − Pr{d(x, y) ≤ rx } y

 s(rx ) ≤ 1 − Pr{d(x, y) = rx } y

1

This definition of rx is not meaningful if we merely show the existence. However, it is used when we consider a deterministic construction. See the next lemma.

8

K. Makino, S. Tamaki, and M. Yamamoto

 s(rx ) ˆ rx −1 ≤ 1 − (3/7)m 2 /( + 1)   ˆ rx −1 ≤ exp −(3/7)m 2 s(rx )/( + 1) ≤ exp (−(2m ˆ − 1)) ˆ Thus, from the union bound, the probability that some x ∈ ({0, 1}3 \ 03 )m is not m ˆ ˆ − 1)) = o(1). Therefore, there covered with any C(i) is at most 7 · exp(−(2m does exist an []-covering code stated in this lemma. 

Note that this lemma only shows the existence of such an []-covering code. We need to deterministically construct it. However, we can get around this issue in the same way as [5]: applying the approximation algorithm for the set cover problem. But, since an []-covering code is not of uniform radius, we can not directly apply the approximation algorithm. ˆ Let Lemma 4. Let d ≥ 2 be a constant that divides m, ˆ and let m ˆ  = m/d.  ˆ  be the maximum integer such that (3/7)m < (1/2) −2 . Let  =  d. Then, there is a polynomial qd (m) ˆ that satisfies the following: an []-covering code ˆ ˆ such that |C(i)| ≤ qd (m) ˆ · (7/3)m /2i C(0), C(1), . . . , C() for ({0, 1}3 \ 03 )m ˆ for 0 ≤ i ≤ , can be deterministically constructed in time poly(m) ˆ · 73m/d + ˆ ˆ · (7/3)m . qd (m) 

ˆ Proof. Let si = 8m ˆ 2 (7/3)m /2i for each i : 0 ≤ i ≤  . First, we deterministically  ˆ such construct an [ ]-covering code D (0), D (1), . . . , D ( ) for ({0, 1}3 \ 03 )m that |D (i)| ≤ poly(m ˆ  ) · si . (Then, we concatenate all of them. See below for details.) Recall the proof of the previous lemma: Let pi = (1/2)i Pr{d(x, y) = i} for ˆ each i : 0 ≤ i ≤  . For any x ∈ ({0, 1}3 \03 )m , we have defined rx = arg max{pi : 0 ≤ i ≤  }, which depends only on x. Note here that rx is not random but a fixed value. Then, we have concluded that the sequence C  (0), C  (1), . . . , C  ( ) of random codes of |C  (i)| ≤ si satisfies the following with high probability: every ˆ is covered with the random code C  (rx ). We can regard this x ∈ ({0, 1}3 \ 03 )m ˆ fact as follows: Let [A0 , A1 , . . . , A ] be a partition of ({0, 1}3 \ 03 )m , where   def ˆ Ai = x ∈ ({0, 1}3 \ 03 )m : rx = i .

Then, there exists an [ ]-covering code C  (0), C  (1), . . . , C  ( ) of |C  (i)| ≤ si such that C  (i) covers Ai . The point of the proof is that we apply the approximation algorithm for the ˆ ), from set cover problem to each Ai (not to the whole space ({0, 1}3 \ 03 )m which we (deterministically) obtain a covering code for each Ai . For this, we obtain all elements of Ai and keep them. This is done by calculating the value ˆ of rx for each x ∈ ({0, 1}3 \ 03 )m . Furthermore, the calculation of rx is done by ˆ calculating pj for every j : 0 ≤ j ≤  : enumerate all y ∈ ({0, 1}3 \ 03 )m such that d(x, y) = j, and then calculate the probability that y is generated by the function init-assign. Then, summing up those values of the probability, we can calculate Pr{d(x, y) = j}, and hence pj . Choosing j as rx such that pj is

Derandomizing HSSW Algorithm for 3-SAT

9

the maximum of all j : 0 ≤ j ≤  , we can obtain the value of rx , and hence Ai . ˆ In total, it takes poly(m) ˆ · 72m time for that procedure. Now, we apply the approximation algorithm for the set cover problem to each Ai . (The approximation algorithmtakes {B(z, i) ∩ Ai : z ∈ Ai } as input, and outputs a set S ⊂ Ai such that z∈S B(z, i) ∩ Ai = Ai .) As is similar to [5], the approximation algorithm finds a covering code D (i) for Ai such that |D (i)| ≤ poly(m ˆ  )·si because at least C  (i) of size at most si covers Ai , and hence the size of an optimal covering code for Ai is also at most si . Furthermore, this ˆ ˆ , it takes poly(m ˆ  ) · 73m is done in time poly(m ˆ  ) · |Ai |3 . In total, since |Ai | ≤ 7m time for that procedure. So far, we have obtained an [ ]-covering code D (0), D (1), . . . , D ( ) for ˆ ({0, 1}3 \ 03 )m such that |D (i)| ≤ poly(m ˆ  ) · si . For each 0 ≤ i ≤  =  d, let C(i) = {D (i1 ) × D (i2 ) × · · · × D (id ) : i = i1 + i2 + · · · + id , 0 ≤ ij ≤  }. def

ˆ It is easy to see that C(0), C(1), . . . , C() is an []-covering code for ({0, 1}3 \03 )m . We (naively) estimate the upper bound on |C(i)|. Let i1 , i2 , · · · , id be integers such that i = i1 + i2 + · · · + id and 0 ≤ ij ≤  . Then,

|D (i1 ) × D (i2 ) × · · · × D (id )| 



ˆ ˆ ˆ 8m ˆ 2 (7/3)m 8m ˆ 2 (7/3)m 8m ˆ 2 (7/3)m = (poly(m ˆ )) · · · · · · · 2i1 2i2 2id 2 d m ˆ d (8m ˆ ) (7/3) = (poly(m ˆ  ))d · 2i1 +···+id ˆ (7/3)m = (poly(m ˆ  ))d · . 2i 



d

Since the number of combinations i1 , . . . , id such that i = i1 + · · · + id and 0 ≤ ij ≤  is at most ( + 1)d , we have ˆ ) · |C(i)| ≤ ( + 1)d · poly(m

ˆ ˆ (7/3)m (7/3)m ≤ qd (m) ˆ · i i 2 2

for some polynomial qd (m). ˆ Finally, we check the running time needed to construct C(i). It takes poly(m) ˆ · ˆ 73m/d time to construct the [ ]-covering code D (0), D (1), . . . , D ( ) for ({0, 1}3\  ˆ 03 )m . Furthermore, it takes i=0 |C(i)| time to construct the []-covering code ˆ m ˆ , which is at most qd (m)·(7/3) ˆ . Summing C(0), C(1), . . . , C() for ({0, 1}3 \03 )m 3m/d ˆ m ˆ up, it takes poly(m) ˆ ·7 + qd (m) ˆ · (7/3) in total.  Recall that |ϕ | = m ˆ = Ω(n). Let n = n − 3m, ˆ which is the number of variables  in ϕ not appeared in ϕ . For the space {0, 1}n , we use an ordinary covering code, that is guaranteed by Lemma 2 to be deterministically constructed. Corollary 1. Let d be a sufficiently large positive constant, and let 0 < ρ < 1/2. Then, there is a polynomial qd (n) that satisfies the following: an {i + ρn :

10

K. Makino, S. Tamaki, and M. Yamamoto

0 ≤ i ≤ }-covering code C(0 + ρn ), C(1 + ρn ), C(2 + ρn ), . . . , C( + ρn ) for  ˆ ˆ (1−h(ρ))n ({0, 1}3 \ 03 )m × {0, 1}n such that |C(i)| ≤ qd (n)(7/3)m 2 /2i , can be m ˆ (1−h(ρ))n deterministically constructed in time qd (n)(7/3) 2 . Proof. It is derived from the previous lemma and Lemma 2. Given an []-covering ˆ , and a ρn -covering code C2 (ρn ) code C1 (0), C1 (1), . . . , C1 () for ({0, 1}3 \ 03 )m n for {0, 1} . For each 0 ≤ i ≤ , let C(i + ρn ) = C1 (i) × C2 (ρn ). def

It is easy to see that C(0 + ρn ), C(1 + ρn ), C(2 + ρn ), . . . , C( + ρn ) is an  ˆ × {0, 1}n . Fur{i + ρn : 0 ≤ i ≤ }-covering code for the space ({0, 1}3 \ 03 )m  ˆ (1−h(ρ))n thermore, |C(i + ρn )| ≤ qd (n)(7/3)m 2 /2i for each i : 0 ≤ i ≤ . From the previous lemma, if the constant d is sufficiently large, the running time for m ˆ ˆ . (deterministically) constructing C1 (0), C1 (1), . . . , C1 () is at most qd (m)(7/3) Similarly, from Lemma 2, the running time for (deterministically) constructing  C2 (ρn ) is at most qd (n )2(1−h(ρ))n . Thus, the total running time is at most 

m ˆ ˆ + qd (n )2(1−h(ρ))n + qd (m)(7/3)

 



ˆ (1−h(ρ))n qd (n)(7/3)m 2 /2i

i=0 

ˆ (1−h(ρ))n ≤ qd (n)(7/3)m 2

for some polynomial qd (n).



Now, using this corollary, we show a derandomization of HSSW, and hence we prove Theorem 1. The outline of the deterministic algorithm is almost same as HSSW, which is described in Fig. 1. We show the derandomization for the case of |ϕ | > αn. Given ϕ , we deterministically construct an {i + ρn : 0 ≤ i ≤ }covering code C(0 + ρn ), C(1 + ρn ), C(2 + ρn ), . . . , C( + ρn ), as is specified in the proofs of Lemma 2, Lemma 4, and Corollary 1. For any z ∈ {0, 1}n and non-negative integer i, we denote by B(z, i) the set of w ∈ {0, 1}n such that d(z, w) ≤ i. Then, given such an {i + ρn : 0 ≤ i ≤ }-covering code, we check whether there is a satisfying assignment within B(z, i + ρn ) for each 0 ≤ i ≤  and each z ∈ C(i + ρn ). It is easy to see that this algorithm finds a satisfying assignment of ϕ if and only if ϕ is satisfiable. We estimate the running time of the algorithm. For any fixed i and z, the  search of a satisfying assignment within B(z, i + ρn ) is done in time (2 + )i+ρn for any small constant  > 0, which is guaranteed by Theorem 2. Thus, given an {i + ρn : 0 ≤ i ≤ }-covering code, the running time for this task for all B(z, i + ρn ) is at most  ˆ   (7/3)m  (1−h(ρ))n qd (n) · · 2 · 2i+ρn · (1 + )n i 2 0≤i≤  m ˆ     7 = qd (n) · · 2(1−h(ρ))n · 2ρn · (1 + )n 3

Derandomizing HSSW Algorithm for 3-SAT

11

 m ˆ  n 7 4 = qd (n) · · · (1 + )n (∵ ρ = 1/4) 3 3  n  m ˆ 4 63 = qd (n) · · · (1 + )n , (∵ n = n − 3m) ˆ 3 64 for some polynomial qd (n). Note from the above corollary that the running time for constructing {i + ρn : 0 ≤ i ≤ }-covering code is less than the above value. Thus, the total running time in case of |ϕ | > αn is at most n ˆ  O((4/3) (63/64)m (1 + )n ) for any  > 0. (Compare this value with the success probability of (2).) On the other hand, it is easy to see that the running ˆ  m ). Therefore, by setting α so that time in case of |ϕ | ≤ αn is at most O(7 n αn n αn (4/3) (63/64) (1 + ) = 7 holds (with  > 0 arbitrarily small), we obtain n  ). the running time O(1.3303

4

Conclusion

We have shown a full derandomization of HSSW, and thereby present a currently fastest deterministic algorithm for 3-SAT. An obvious future work is to obtain a full derandomization of the currently best known randomized algorithm for 3-SAT [7]. To do so, it seems to be required to derandomize Paturi et al.’s algorithm [15] completely. Another possible future work is to extend HSSW algorithm to the k-SAT case. It leads to the fastest deterministic algorithms for k-SAT, combined with the derandomization techniques of this paper and Moser and Scheder [13]. Acknowledgements. We are grateful to anonymous referees for giving us useful comments for improving the presentation.

References 1. Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Annals of Mathematics 160(2), 781–793 (2004) 2. Baumer, S., Schuler, R.: Improving a probabilistic 3-SAT algorithm by dynamic search and independent clause pairs. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 150–161. Springer, Heidelberg (2004) 3. Br¨ ueggemann, T., Kern, W.: An improved deterministic local search algorithm for 3-SAT. Theoretical Computer Science 329(1-3), 303–313 (2004) 4. Chandrasekaran, K., Goyal, N., Haeupler, B.: Deterministic Algorithms for the Lov´ asz Local Lemma. In: Proc. of SODA 2010, pp. 992–1004 (2010) 5. Dantsin, E., Goerdt, A., Hirsch, E., Kannan, R., Kleinberg, J., Papadimitriou, C., Raghavan, P., Sch¨ oning, U.: A deterministic (2 − 2/(k + 1))n algorithm for k-SAT based on local search. Theoretical Computer Science 289(1), 69–83 (2002) 6. Dantsin, E., Hirsch, E.A.: Worst-Case Upper Bounds. In: Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 403–424. IOS Press, Amsterdam (2009)

12

K. Makino, S. Tamaki, and M. Yamamoto

7. Hertli, T., Moser, R.A., Scheder, D.: Improving PPSZ for 3-SAT using Crititical Variables. In: Proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS), pp. 237–248 (2011) 8. Hofmeister, T., Sch¨ oning, U., Schuler, R., Watanabe, O.: A probabilistic 3-SAT algorithm further improved. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 192–202. Springer, Heidelberg (2002) 9. Iwama, K., Seto, K., Takai, T., Tamaki, S.: Improved Randomized Algorithms for 3-SAT. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010. LNCS, vol. 6506, pp. 73–84. Springer, Heidelberg (2010) 10. Iwama, K., Tamaki, S.: Improved upper bounds for 3-SAT. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 328– 329 (2004) 11. Kutzkov, K., Scheder, D.: Using CSP To Improve Deterministic 3-SAT. arXiv:1007.1166v2 (2010) 12. Monien, B., Speckenmeyer, E.: Solving satisfiability in less than 2n steps. Discrete Applied Mathematics 10, 287–295 (1985) 13. Moser, R., Scheder, D.: A Full Derandomization of Sch¨ oning’s k-SAT Algorithm. In: Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC), pp. 245–252 (2011) arXiv:1008.4067v1 14. Mahajan, S., Ramesh, H.: Derandomizing Approximation Algorithms Based on Semidefinite Programming. SIAM J. Comput. 28(5), 1641–1663 (1999) 15. Paturi, R., Pudl´ ak, P., Saks, M., Zane, F.: An Improve Exponential-Time Algorithm for k-SAT. In: Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 628–637 (1998); Journal version: J. of the ACM 52(3), 337–364 (2005) 16. Paturi, R., Pudl´ ak, P., Zane, F.: Satisfiability coding lemma. In: Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 566–574 (1997) 17. Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4), Article 17 (2008) 18. Rolf, D.: 3-SAT ∈ RTIME(O(1.32793n )). Electronic Colloquium on Computational Complexity, TR03-054 (2003) 19. Rolf, D.: Improved bound for the PPSZ/Sch¨ oning-algorithm for 3-SAT. Journal on Satisfiability, Boolean Modeling and Computation 1, 111–122 (2006) 20. Sch¨ oning, U.: A probabilistic algorithm for k-SAT and constraint satisfaction problems. In: Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 410–414 (1999) 21. Scheder, D.: Guided search and a faster deterministic algorithm for 3-SAT. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 60–71. Springer, Heidelberg (2008)

Dominating Set Counting in Graph Classes Shuji Kijima1 , Yoshio Okamoto2, and Takeaki Uno3 1

Graduate School of Information Science and Electrical Engineering, Kyushu University, Japan [email protected] 2 Center for Graduate Education Initiative, Japan Advanced Institute of Science and Technology, Japan [email protected] 3 National Institute of Informatics, Japan [email protected]

Abstract. We make an attempt to understand the dominating set counting problem in graph classes from the viewpoint of polynomial-time computability. We give polynomial-time algorithms to count the number of dominating sets (and minimum dominating sets) in interval graphs and trapezoid graphs. They are based on dynamic programming. With the help of dynamic update on a binary tree, we further reduce the time complexity. On the other hand, we prove that counting the number of dominating sets (and minimum dominating sets) in split graphs and chordal bipartite graphs is #P-complete. These results are in vivid contrast with the recent results on counting the independent sets and the matchings in chordal graphs and chordal bipartite graphs.

1 Introduction Combinatorics is a branch of mathematics that often deals with counting various objects, and has a long tradition. However, the algorithmic aspect of counting has been less studied. This seems due to the facts that most of the problems turn out to be #Phard (thus unlikely to have polynomial-time algorithms) and that not many algorithmic techniques have been known. In the study of graph classes, the situation does not differ. There are many studies on decision problems and optimization problems, but fewer studies on counting problems. Certainly, counting algorithms require properties of graphs that are not needed for solving decision and optimization problems. From this perspective, Okamoto, Uehara and Uno studied two basic counting problems for graph classes. The first paper [8] studied the problem to count the number of independent sets, and provided a linear-time algorithm for chordal graphs. On the other hand, their second paper [7] in the series studied the problems to count the number of matchings and perfect matchings, respectively, and proved that the problem is #P-complete for chordal graphs (actually for split graphs). They are also #P-complete for chordal bipartite graphs. It still remains open whether 

The first and second authors are supported by Grant-in-Aid for Scientific Research from Ministry of Education, Science and Culture, Japan, and Japan Society for the Promotion of Science.

B. Fu and D.-Z. Du (Eds.): COCOON 2011, LNCS 6842, pp. 13–24, 2011. c Springer-Verlag Berlin Heidelberg 2011 

14

S. Kijima, Y. Okamoto, and T. Uno

AT-free

weakly chordal

bipartite

cocomparability

chordal

chordal bipartite

cobipartite permutation

trapezoid

strongly chordal interval

split

Fig. 1. Complexity landscape for the dominating set counting problem regarding graph classes

the number of matchings (or perfect matchings) can be computed in polynomial time for interval graphs, and even for proper interval graphs. This paper is concerned with dominating sets. In this paper, we will try to understand the dominating set counting problem from the viewpoint of polynomial-time computability. Domination is one of the main subjects in graph theory and graph algorithms, and there are some books especially devoted to that topic [3,4]. However, not much is known for the counting aspect. Our result. We give polynomial-time algorithms to count the number of dominating sets (and minimum dominating sets) in interval graphs and trapezoid graphs. They are based on dynamic programming. With the help of dynamic update on a binary tree, we reduce the time complexity. On the other hand, we prove that counting the number of dominating sets in split graphs, chordal bipartite graphs, and cobipartite graphs is #P-complete, and counting the number of minimum dominating sets in split graphs and chordal bipartite graphs is #P-hard. Fig. 1 summarizes our results on counting the number of dominating sets. In the figure, each arrow means that the class on the tail is a subclass of the class on the head. For the graph classes in solid lines polynomial-time algorithms exist, and for the graph classes in double solid lines the problem is #P-complete. For strongly chordal graphs, in a dashed line, the question is left open. As for counting the minimum dominating sets, Kratsch [5] very recently gave an O(n7 )-time algorithm for AT-free graphs, while the polynomial-time computability is open for strongly chordal graphs. Note that a trapezoid graph is AT-free, but the bound of our algorithm is better than Kratsch’s algorithm. Related work. On counting the number of independent dominating sets and the number of minimum independent dominating sets, Okamoto, Uno and Uehara [8] proved the #P-hardness for chordal graphs, and Lin and Chen [6] gave linear-time algorithms for interval graphs. We note that the parity version of the problem turns out to be trivial: Brouwer, Csorba, and Schrijver [2] proved that every graph has an odd number of dominating sets. Preliminaries. In this paper, all graphs are finite, simple and undirected. A graph G is denoted by a pair (V, E) of its vertex set V and its edge set E. The set of adjacent vertices of v is called the neighborhood of v, and denoted by NG (v). If there is no we simply write N (v). For a vertex subset X ⊆ V , we denote N (X) := confusion,  N (v) \ X. v∈X

Dominating Set Counting in Graph Classes

6

2

y1 y2

x1 x2

1

1 3

2

4

5

3 4

An interval graph G

15

6

5

7

An interval representation of G

Fig. 2. Example of an interval graph and the corresponding intervals

For u, v ∈ V , we say u dominates v if u and v are adjacent in G. A subset D ⊆ V dominates v if at least one vertex in D dominates v. A dominating set of G is a vertex subset D ⊆ V that dominates all vertices in V \ D. A dominating set of G is minimum if it has the smallest number of elements among all dominating sets of G. In the analysis of our algorithms, the time complexity refers to the number of arithmetic operations, not bit operations. Due to the space constraint, we postpone the proofs of theorems/lemmas with * marks to the full version.

2 Interval Graphs A graph G = ({1, . . . , n}, E) is interval if there exists a set {I1 , . . . , In } of closed intervals on a real line such that {i, j} ∈ E if and only if Ii ∩ Ij = ∅. Such a set of intervals is called an interval representation of G. It is known that there always exists an interval representation in which the endpoints of intervals are all distinct, and we can find such a representation of a given interval graph that is sorted by their right endpoints in O(n + m) time [1], where n is the number of vertices and m is the number of edges. Let {I1 , . . . , In } be such a representation, and let each interval be represented as Ii = [xi , yi ]. By the assumption, it holds that yi < yj if i < j. For the sake of technical simplicity, in the following we add an extra interval [xn+1 , yn+1 ] satisfying that xn+1 > yi for all i ∈ {1, . . . , n} and xn+1 < yn+1 . See an example in Fig. 2. Note that this addition only doubles the number of dominating sets. 2.1 Counting the Dominating Sets The basic strategy is to apply the dynamic programming. To this end, we look at the intervals from left to right, and keep track of the following families of subsets. Let F (i, j) denote the family of subsets S ⊆ {1, . . . , n} satisfying the following two conditions; 1. j = max{j  ∈ {1, . . . , n} | j  ∈ S}, and 2. i = min{i ∈ {1, . . . , n + 1} | S does not dominate i }. Note that for any S ⊆ {1, . . . , n}, there exists a unique pair (i, j) such that S ∈ F (i, j). Moreover, S ⊆ {1, . . . , n} is a dominating set of the interval graph if and only if S ∈ F (n + 1, j) for some j. Therefore, we readily obtain the following lemma.

16

S. Kijima, Y. Okamoto, and T. Uno

Lemma 1. The number of dominating sets of the interval graph is

n 

|F (n + 1, j)| .

j=1



The next is our key lemma, which indicates an algorithm based on dynamic programming. Let i∗ (j) := min{i | i ∈ N (j) ∪ {j}, i > j}, and let F ⊗ {{j}} denote {S  ∪ {j} | S  ∈ F } for F ⊆ 2{1,...,j−1} . For instance, in the example of Fig. 2, we have i∗ (1) = 3 and i∗ (3) = 7. Lemma 2. For j = 1, – F (i, 1) = {{1}} if i = i∗ (1), and – F (i, 1) = ∅ otherwise. For j ∈ {2, . . . , n + 1}, – F (i, j) = ⎛ ∅ if i ∈ N (j) ∪⎞{j},  – F (i, j) = ⎝ F (i, j  )⎠ ⊗ {{j}} if i ∈ N (j) ∪ {j} and i < j, j  j. If i > i∗ (j), then F (i, j) = ∅ since i∗ (j) is ∗ way to not dominated by j. If i = i∗ (j),   then by the definition of i (j), in a similar    F (i, j  ) ∪ F (i , j  ) ⊗ {{j}} = Case 2, it holds that F (i, j) =    i ∈N (j)∪{j} j