Edition Revised and Updated nd Edited by Vangelis Th. Paschos Optimization Problems and New Approaches 2 Paradigms Paradigms of MATHEMATICS AND STATISTICS SERIES STATISTICS AND MATHEMATICS Combinatorial
Paradigms of Edited by Vangelis Th. Paschos Combinatorial Optimization 2nd Edition Revised and Updated Z(7ib8e8-CBGFHA( is Exceptional Class Professor of Computer Science and www.iste.co.uk This updated and revised Optimization series 2nd covers a very large edition set of topics in of this area, of dealing applications classical several as well as approaches the and notions fundamental with three-volume Combinatorial Combinatorial Optimization. Combinatorial Optimization is a multidisciplinary field, lying at the interface of three major scientific domains: science, and management applied studies. Its focus is on finding the least-cost solution mathematics, theoretical computer to a mathematical the so feasible, not is search exhaustive problems, problem such many In cost. numerical in which each approach solution taken is to is operate within the associated domain of optimization which with the set of feasible solutions is discrete or can be reduced to discrete, and problems, in a involving problems common Some solution. best the find to is goal the which in combinatorial optimization are minimum spanning tree problem. the traveling salesman problem Combinatorial and Optimization the is a operations research, algorithm and theory, computational complexity subset It theory. of optimization has that important is applications related in several to mathematics, and software engineering. fields, including artificial intelligence, This second volume, which addresses the various paradigms and approaches taken in Combinatorial Optimization, is divided into two parts: - Paradigmatic Problems, TSP, satisfiability optimal coloring, min cut, max as such problems, optimization which discusses several etc., famous the combinatorial study of legitimization which and the establishment has of Combinatorial largely Optimization as one contributed of the most active current scientific domains. to the development, the - New Approaches, which presents the methodological approaches that fertilize and are fertilizedapproximation, on-line by computation, Combinatorial robustness, algorithmic game theory. etc., Optimization and, more such recently, The as three volumes of this polynomial series form a coherent whole. intended to The be set a of self-contained books treatment is requiring only basic understanding and knowledge of a few mathematical theories and concepts. It is intended for researchers, practitioners and MSc or PhD students. Vangelis Th. Paschos Combinatorial Optimization at the Paris-Dauphine University and chairman of Aiding Decision of Analysis the and Modeling the for (Laboratory LAMSADE the Systems) in France. W657-Paschos 2.qxp_Layout 1 01/07/2014 14:05 Page 1 14:05 Page 2.qxp_Layout 1 01/07/2014 W657-Paschos
Paradigms of Combinatorial Optimization
Revised and Updated 2nd Edition
Paradigms of Combinatorial Optimization
Problems and New Approaches
Edited by Vangelis Th. Paschos First edition published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.© ISTE Ltd 2010 First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com
© ISTE Ltd 2014 The rights of Vangelis Th. Paschos to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2014942904
British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-657-0
Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY Table of Contents
Preface ...... xvii Vangelis Th. PASCHOS
PART I. PARADIGMATIC PROBLEMS...... 1 Chapter 1. Optimal Satisfiability...... 3 Cristina BAZGAN 1.1. Introduction...... 3 1.2.Preliminaries...... 5 1.2.1. Constraint satisfaction problems: decision and optimization versions ...... 6 1.2.2.Constrainttypes...... 8 1.3.Complexityofdecisionproblems...... 10 1.4.Complexityandapproximationofoptimizationproblems...... 13 1.4.1.Maximizationproblems...... 13 1.4.2.Minimizationproblems...... 20 1.5.Particularinstancesofconstraintsatisfactionproblems...... 20 1.5.1.Planarinstances...... 21 1.5.2.Denseinstances...... 22 1.5.3.Instanceswithaboundednumberofoccurrences...... 24 1.6.Satisfiability problems under global constraints ...... 25 1.7.Conclusion...... 27 1.8.Bibliography...... 27
Chapter 2. Scheduling Problems ...... 33 Philippe CHRÉTIENNE and Christophe PICOULEAU 2.1. Introduction ...... 33 2.2.Newtechniquesforapproximation...... 34 vi Combinatorial Optimization 2
2.2.1. Linear programming and scheduling ...... 35 2.2.2.AnapproximationschemeforP||Cmax...... 40 2.3.Constraintsandscheduling...... 41 2.3.1. The monomachine constraint ...... 41 2.3.2.Thecumulativeconstraint...... 44 2.3.3.Energeticreasoning...... 45 2.4.Non-regularcriteria...... 46 2.4.1.PERTwithconvexcosts...... 47 2.4.2.Minimizingtheearly–tardycostononemachine...... 52 2.5.Bibliography...... 57
Chapter 3. Location Problems ...... 61 Aristotelis GIANNAKOS 3.1. Introduction...... 61 3.1.1.Weber’sproblem...... 62 3.1.2.Aclassification...... 64 3.2.Continuous problems ...... 65 3.2.1.Completecovering...... 65 3.2.2.Maximalcovering...... 66 3.2.3.Emptycovering...... 67 3.2.4.Bicriteriamodels...... 69 3.2.5.Covering with multiple resources ...... 69 3.3.Discreteproblems...... 70 3.3.1.p-Center...... 70 3.3.2.p-Dispersion...... 70 3.3.3.p-Median...... 71 3.3.4.Hub...... 73 3.3.5.p-Maxisum...... 73 3.4. On-line problems ...... 74 3.5.Thequadraticassignmentproblem...... 77 3.5.1.Definitions and formulations of the problem...... 77 3.5.2.Complexity...... 79 3.5.3.Relaxationsandlowerbounds...... 79 3.6.Conclusion...... 82 3.7.Bibliography...... 83
Chapter 4. MiniMax Algorithms and Games ...... 89 Michel KOSKAS 4.1. Introduction...... 89 4.2.Gamesofnochance:thesimplecases...... 91 4.3.Thecaseofcomplexnochancegames...... 94 4.3.1.Approximativeevaluation...... 95 Table of Contents vii
4.3.2. Horizon effect ...... 97 4.3.3. α-β pruning ...... 97 4.4.Quiescencesearch...... 99 4.4.1.OtherrefinementsoftheMiniMaxalgorithm...... 102 4.5.Caseofgamesusingchance...... 103 4.6.Conclusion...... 103 4.7.Bibliography...... 106
Chapter 5. Two-dimensional Bin Packing Problems ...... 107 Andrea LODI, Silvano MARTELLO, Michele MONACI and Daniele VIGO 5.1. Introduction...... 107 5.2.Models...... 108 5.2.1.ILPmodelsforlevelpacking...... 109 5.3.Upperbounds...... 112 5.3.1.Strippacking...... 112 5.3.2.Binpacking:two-phaseheuristics...... 113 5.3.3.Binpacking:one-phaselevelheuristics...... 115 5.3.4.Binpacking:one-phasenon-levelheuristics...... 116 5.3.5.Metaheuristics...... 116 5.3.6.Approximationalgorithms...... 118 5.4.Lowerbounds...... 119 5.4.1.Lowerboundsforlevelpacking...... 123 5.5.Exactalgorithms...... 123 5.6Acknowledgements...... 125 5.7.Bibliography...... 125
Chapter 6. The Maximum Cut Problem ...... 131 Walid BEN-AMEUR, Ali Ridha MAHJOUB and José NETO 6.1. Introduction...... 131 6.2.Complexityandpolynomialcases...... 133 6.3.Applications...... 134 6.3.1.Spinglassmodels...... 134 6.3.2.Unconstrained0–1quadraticprogramming...... 135 6.3.3.Theviaminimizationproblem...... 136 6.4.Thecutpolytope...... 137 6.4.1.Valid inequalities and separation...... 137 6.4.2.Branch-and-cutalgorithms...... 142 6.4.3.Thecutpolyhedron...... 144 6.5.Semi-definiteprogramming(SDP)andthemaximumcutproblem... 145 6.5.1.Semi-definiteformulationoftheMAX-CUTproblem...... 146 6.5.2.Quality of the semi-definite formulation ...... 147 6.5.3.Existing works in the literature ...... 150 viii Combinatorial Optimization 2
6.6. The cut cone and applications ...... 152 6.6.1.Thecutcone...... 152 6.6.2.Relationshiptothecutpolytope...... 152 6.6.3.Thesemi-metriccone...... 153 6.6.4.Applications to the multiflow problem ...... 155 6.7. Approximation methods ...... 157 6.7.1.Methodswithperformanceguarantees...... 157 6.7.2.Methodswithnoguarantees...... 158 6.8.Relatedproblems...... 159 6.8.1.Unconstrained0–1quadraticprogramming...... 159 6.8.2.Themaximumeven(odd)cutproblem...... 160 6.8.3.The equipartition problem...... 161 6.8.4.Otherproblems...... 162 6.9.Conclusion...... 163 6.10.Bibliography...... 164
Chapter 7. The Traveling Salesman Problem and its Variations ...... 173 Jérôme MONNOT and Sophie TOULOUSE 7.1. Introduction...... 173 7.2.Elementarypropertiesandvarioussubproblems...... 174 7.2.1.Elementaryproperties...... 174 7.2.2.Varioussubproblems...... 175 7.3.Exactsolvingalgorithms...... 177 7.3.1.Adynamicprogrammingalgorithm...... 177 7.3.2.Abranch-and-boundalgorithm...... 179 7.4.ApproximationalgorithmformaxTSP...... 184 7.4.1.Analgorithmbasedon2-matching...... 186 7.4.2.Algorithmmixing2-matchingandmatching...... 189 7.5.ApproximationalgorithmforminTSP...... 192 7.5.1.Algorithmbasedonthespanningtreeandmatching...... 196 7.5.2.Localsearchalgorithm...... 197 7.6.Constructivealgorithms...... 201 7.6.1.Nearestneighboralgorithm...... 201 7.6.2.Nearestinsertionalgorithm...... 207 7.7.Conclusion...... 210 7.8.Bibliography...... 211
Chapter 8. 0–1 Knapsack Problems ...... 215 Gérard PLATEAU and Anass NAGIH 8.1. General solution principle ...... 215 8.2.Relaxation...... 217 8.3.Heuristic...... 222 Table of Contents ix
8.4. Variable fixing ...... 222 8.5.Dynamicprogramming...... 226 8.5.1.Generalprinciple...... 227 8.5.2.Managingfeasiblecombinationsofobjects...... 230 8.6. Solution search by hybridization of branch-and-bound and dynamic programming ...... 234 8.6.1.Principleofhybridization...... 235 8.6.2.Illustrationofhybridization...... 237 8.7.Conclusion...... 239 8.8.Bibliography...... 240
Chapter 9. Integer Quadratic Knapsack Problems ...... 243 Dominique QUADRI, Eric SOUTIF and Pierre TOLLA 9.1. Introduction...... 243 9.1.1.Problemformulation...... 243 9.1.2.Significanceoftheproblem...... 244 9.2. Solution methods ...... 246 9.2.1.Theconvexseparableproblem...... 246 9.2.2.Thenon-convexseparableproblem...... 252 9.2.3.Theconvexnon-separableproblem...... 254 9.2.4.Thenon-convexnon-separableproblem...... 256 9.3.Numericalexperiments...... 259 9.3.1.Theconvexandseparableintegerquadraticknapsackproblem... 260 9.3.2.The convex and separable integer quadratic multi-knapsack problem ...... 260 9.4.Conclusion...... 261 9.5.Bibliography...... 261
Chapter 10. Graph Coloring Problems ...... 265 Dominique DE WERRA and Daniel KOBLER 10.1. Basic notions of colorings ...... 265 10.2.Complexityofcoloring...... 269 10.3. Sequential methods of coloring ...... 270 10.4.Anexactcoloringalgorithm...... 272 10.5.Tabusearch...... 276 10.6.Perfectgraphs...... 280 10.7.Chromaticscheduling...... 285 10.8.Intervalcoloring...... 287 10.9. T-colorings ...... 289 10.10.Listcolorings...... 292 10.11. Coloring with cardinality constraints ...... 295 10.12.Otherextensions...... 298 x Combinatorial Optimization 2
10.13. Edge coloring ...... 299 10.13.1. f-Coloring of edge ...... 300 10.13.2. [g, f]-Colorings of edges ...... 301 10.13.3.Amodelofhypergraphcoloring...... 303 10.14.Conclusion...... 306 10.15.Bibliography...... 307
PART II. NEW APPROACHES ...... 311
Chapter 11. Polynomial Approximation ...... 313 Marc DEMANGE and Vangelis Th. PASCHOS 11.1. What is polynomial approximation? ...... 313 11.1.1.Efficientlysolvingadifficultproblem...... 314 11.1.2.Approximationmeasures...... 314 11.2.Somefirstexamplesofanalysis:constantapproximationratios.... 316 11.2.1. An example of classical approximation: the metric traveling salesman...... 316 11.2.2.Examplesofthedifferentialratiocase...... 317 11.3.Approximationschemes...... 323 11.3.1.Non-completeschemes...... 323 11.3.2. Complete approximation schemes – example of the Boolean knapsack ...... 333 11.4.Analysesdependingontheinstance...... 336 11.4.1.Setcoveringandclassicalratios...... 336 11.4.2.Setcoveringanddifferentialratios...... 337 11.4.3.Themaximumstablesetproblem...... 338 11.5.Conclusion:methodsandissuesofapproximation...... 339 11.5.1.Thetypesofalgorithms:afewgreatclassics...... 340 11.5.2.Approximationclasses:structuringtheclassNPO...... 341 11.5.3.Reductionsinapproximation...... 344 11.5.4.Issues...... 345 11.6.Bibliography...... 346
Chapter 12. Approximation Preserving Reductions ...... 351 Giorgio AUSIELLO and Vangelis Th. PASCHOS 12.1. Introduction ...... 351 12.2.Strictandcontinuousreductions...... 353 12.2.1.Strictreductions...... 353 12.2.2.Continuousreduction...... 357 12.3. AP-reduction and completeness in the classes NPO and APX ..... 359 12.3.1.CompletenessinNPO...... 360 Table of Contents xi
12.3.2. Completeness in APX ...... 362 12.3.3.Usingcompletenesstoderivenegativeresults...... 365 12.4. L-reduction and completeness in the classes Max-SNP and APX ... 366 12.4.1. The L-reduction and the class Max-SNP...... 366 12.4.2.ExamplesofL-reductions...... 367 12.4.3.CompletenessinMax-SNPandAPX...... 370 12.5.Affinereduction...... 371 12.6. A few words on GAP-reduction ...... 373 12.7.Historyandcomment...... 374 12.8.Bibliography...... 378
Chapter 13. Inapproximability of Combinatorial Optimization Problems ...... 381 Luca TREVISAN 13.1. Introduction ...... 381 13.1.1.Abriefhistoricaloverview...... 382 13.1.2.Organizationofthischapter...... 385 13.1.3.Furtherreading...... 386 13.2.Sometechnicalpreliminaries...... 387 13.3. Probabilistically checkable proofs...... 389 13.3.1. PCP and the approximability of Max SAT ...... 390 13.4.Basicreductions...... 392 13.4.1. Max 3SAT with bounded occurrences...... 392 13.4.2.VertexCoverandIndependentSet...... 394 13.4.3.Steinertreeproblem...... 396 13.4.4.MoreaboutIndependentSet...... 398 13.5.OptimizedreductionsandPCPconstructions...... 400 13.5.1.PCPsoptimizedforMaxSATandMaxCUT...... 400 13.5.2.PCPsoptimizedforIndependentSet...... 402 13.5.3.TheUniqueGamesConjecture...... 403 13.6.Anoverviewofknowninapproximabilityresults...... 404 13.6.1.Latticeproblems...... 404 13.6.2.Decodinglinearerror-correctingcodes...... 406 13.6.3.Thetravelingsalesmanproblem...... 407 13.6.4.Coloringproblems...... 409 13.6.5.Coveringproblems...... 409 13.6.6. Graph partitioning problems...... 411 13.7. Integrality gap results ...... 412 13.7.1.ConvexrelaxationsoftheSparsestCutproblem...... 413 13.7.2. Families of relaxations ...... 413 13.7.3.IntegralitygapsversusUniqueGames...... 415 13.8.Othertopics...... 416 xii Combinatorial Optimization 2
13.8.1. Complexity classes of optimization problems ...... 416 13.8.2.Average-casecomplexityandapproximability...... 418 13.8.3.WitnesslengthinPCPconstructions...... 419 13.8.4.Typicalandunusualapproximationfactors...... 419 13.9.Conclusions...... 421 13.10.Proveoptimalresultsfor2-queryPCPs...... 422 13.11. Settle the Unique Games Conjecture ...... 422 13.12.ProveastronginapproximabilityresultforMetricTSP...... 422 13.13.Acknowledgements...... 423 13.14.Bibliography...... 423
Chapter 14. Local Search: Complexity and Approximation...... 435 Eric ANGEL, Petros CHRISTOPOULOS and Vassilis ZISSIMOPOULOS 14.1. Introduction ...... 435 14.2.Formalframework...... 437 14.3.Afewfamiliaroptimizationproblemsandtheirneighborhoods.... 439 14.3.1. Graph partitioning problem ...... 439 14.3.2.Themaximumcutproblem...... 439 14.3.3.Thetravelingsalesmanproblem...... 440 14.3.4.Clausesatisfactionproblems...... 441 14.3.5.StableconfigurationsinaHopfield-typeneuralnetwork...... 441 14.4.ThePLSclass...... 442 14.5.Complexityofthestandardlocalsearchalgorithm...... 447 14.6. The quality of local optima ...... 449 14.7.Approximationresults...... 450 14.7.1.TheMAXk-SATproblem...... 451 14.7.2.TheMAXCUTproblem...... 452 14.7.3.Otherproblemsongraphs...... 454 14.7.4.Thetravelingsalesmanproblem...... 456 14.7.5.Thequadraticassignmentproblem...... 457 14.7.6.Classificationproblems...... 460 14.7.7. Facility location problems ...... 462 14.8.Conclusionandopenproblems...... 465 14.9.Bibliography...... 467
Chapter 15. On-line Algorithms ...... 473 Giorgio AUSIELLO and Luca BECCHETTI 15.1. Introduction ...... 473 15.2.Someclassicalon-lineproblem...... 475 15.2.1.Listupdating...... 476 15.2.2.Paging...... 477 15.2.3.Thetravelingsalesmanproblem...... 480 Table of Contents xiii
15.2.4. Load balancing ...... 482 15.3.Competitiveanalysisofdeterministicalgorithms...... 483 15.3.1. Competitive analysis of list updating ...... 484 15.3.2. Competitive analysis of paging algorithms ...... 486 15.3.3. Competitive analysis of on-line TSP...... 488 15.3.4. Competitive analysis of on-line load balancing ...... 494 15.4.Randomization...... 496 15.4.1.Randomizedpaging...... 497 15.4.2. Lower bounds: Yao’s lemma and its application to paging algorithm ...... 499 15.5. Extensions of competitive analysis ...... 501 15.5.1.Adhoctechniques:thecaseofpaging...... 502 15.5.2.Generaltechniques...... 503 15.6.Bibliography...... 505
Chapter 16. Polynomial Approximation for Multicriteria Combinatorial Optimization Problems ...... 511 Eric ANGEL, Evripidis BAMPIS and Laurent GOURVÈS 16.1. Introduction ...... 511 16.2. Presentation of multicriteria combinatorial problems ...... 513 16.2.1. Multicriteria combinatorial problems ...... 513 16.2.2. Optimality ...... 514 16.2.3. Complexity of multicriteria combinatorial problems ...... 517 16.3.Polynomialapproximationandperformanceguarantee...... 521 16.3.1.Criteriaweightingapproach...... 521 16.3.2.Simultaneousapproach...... 524 16.3.3.Budgetapproach...... 527 16.3.4.Paretocurveapproach...... 531 16.4.Bibliographicalnotes...... 541 16.4.1. Presentation of multicriteria combinatorial problems ...... 541 16.4.2. Polynomial approximation with performance guarantees ..... 541 16.5.Conclusion...... 543 16.6.Bibliography...... 543
Chapter 17. An Introduction to Inverse Combinatorial Problems ...... 547 Marc DEMANGE and Jérôme MONNOT 17.1. Introduction ...... 547 17.2.Definitionsandnotation...... 549 17.3.Polynomialinverseproblemsandsolutiontechniques...... 552 17.3.1.Thelinearprogrammingcase...... 553 17.3.2.Inversemaximumflowproblem...... 562 17.3.3.Aclassofpolynomialinverseproblems...... 564 xiv Combinatorial Optimization 2
17.3.4. Avenues to explore: the example of the inverse bivalent variable maximum weight matching problem ...... 567 17.4.Hardinverseproblems...... 569 17.4.1.InverseNP-hardproblems...... 569 17.4.2. Facility location problem ...... 572 17.4.3.Apartialinverseproblem:theminimumcapacitycut...... 575 17.4.4.Maximumweightmatchingproblem...... 578 17.5.Conclusion...... 583 17.6.Bibliography...... 584
Chapter 18. Probabilistic Combinatorial Optimization...... 587 Cécile MURAT and Vangelis Th. PASCHOS 18.1. Motivations and applications...... 587 18.2. The issues: formalism and methodology ...... 589 18.3.Problemcomplexity...... 593 18.3.1.MembershipofNPisnotgiven...... 593 18.3.2. Links between the deterministic and probabilistic frameworks from the complexity point of view...... 599 18.4.Solvingproblems...... 601 18.4.1.Characterizationofoptimalsolutions...... 602 18.4.2. Polynomial solution of certain instances ...... 605 18.4.3.Effectivesolution...... 607 18.5.Approximation...... 608 18.6.Bibliography...... 611
Chapter 19. Robust Shortest Path Problems ...... 615 Virginie GABREL and Cécile MURAT 19.1. Introduction ...... 615 19.2. Taking uncertainty into account: the various models...... 616 19.2.1.Theintervalmodel...... 617 19.2.2.Thediscretescenariomode...... 617 19.3.Measuresofrobustness...... 619 19.3.1.Classicalcriterionderivedfromdecision-makingtheory...... 619 19.3.2. Methodology inspired by mathematical programming ...... 622 19.3.3. Methodology inspired by multicriteria analysis ...... 623 19.4. Complexity and solution of robust shortest path problems in the interval mode ...... 625 19.4.1.Withtheworst-casecriterion...... 625 19.4.2.Withthemaximumregretcriterion...... 626 19.4.3.Withthemathematicalprogramminginspiredapproach...... 630 19.4.4.Withthemulticriteriaanalysisinspiredapproach...... 632 Table of Contents xv
19.5. Complexity and solution of robust shortest path problems in a discrete set of scenarios model...... 635 19.5.1.Withtheworst-casecriterion...... 635 19.5.2.Withthemaximumregretcriterion...... 636 19.5.3.Withthemulticriteriaanalysisinspiredapproach...... 637 19.6.Conclusion...... 637 19.7.Bibliography...... 638
Chapter 20. Algorithmic Games ...... 641 Aristotelis GIANNAKOS and Vangelis PASCHOS 20.1. Preliminaries ...... 642 20.1.1.Basicnotionsofgames...... 642 20.1.2.Theclassesofcomplexitycoveredinthischapter...... 645 20.2. Nash equilibria ...... 647 20.3. Mixed extension of a game and Nash equilibria ...... 649 20.4.Algorithmicproblems...... 650 20.4.1.Succinctdescriptiongame...... 651 20.4.2. Results on the complexity of computing a mixed equilibrium... 651 20.4.3. Counting the number of equilibria in a mixed strategy game ... 657 20.5.Potentialgames...... 657 20.5.1.Definitions...... 657 20.5.2.Properties...... 658 20.6.Congestiongames...... 662 20.6.1.Rosenthal’smodel...... 662 20.6.2.Complexityofcongestiongames(Rosenthal’smodel)...... 665 20.6.3.Othermodels...... 666 20.7.Finalnotes...... 670 20.8.Bibliography...... 670 Chapter 21 Combinatorial Optimization with Competing Agents ...... 675 Diodato FERRAIOLI, Laurent GOURVÈS, Stefano MORETTI, Fanny PASCUAl and Olivier SPANJAARD 21.1. Introduction ...... 675 21.2.SocialwelfareoptimizationinSocialNetworks...... 678 21.2.1.Localinteractionmodel...... 679 21.3.Asimpleconnectiongame...... 688 21.3.1.Themodel...... 689 21.3.2.TheBirdrule...... 691 21.3.3.Optimalandbudgetbalancedprotocols...... 692 21.4.Efficiencyoftruthfulalgorithmswithoutpayment...... 695 21.4.1. Illustrative examples of positive and negative results ...... 696 xvi Combinatorial Optimization 2
21.4.2. Ways to go beyond negative results ...... 698 21.4.3. Spotlight results on the facility location problem ...... 699 21.4.4.Spotlightresultsontheallocationproblem...... 701 21.5.Bibliography...... 704
General Bibliography ...... 707 List of Authors ...... 767 Index ...... 773 Summary of Other Volumes in the Series ...... 781 *
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,%#*'0 & 2,# PART I
Paradigmatic Problems
Chapter 1
Optimal Satisfiability
1.1. Introduction
Given a set of constraints defined on Boolean variables, a satisfiability problem, also called a Boolean constraint satisfaction problem, consists of deciding whether there exists an assignment of values to the variables that satisfies all the constraints (and possibly establishing such an assignment). Often, such an assignment does not exist and, in this case, it is natural to seek an assignment that satisfies a maximum number of constraints or minimizes the number of non-satisfied constraints.
An example of a Boolean constraint satisfaction problem is the problem known as SAT, which consists of deciding whether a propositional formula (expressed as a conjunction of disjunctions) is satisfiable or not. SAT was the first problem shown to be NP-complete by Cook [COO 71] and Levin [LEV 73] and it has remained a central problem in the study of NP-hardness of optimization problems [GAR 79]. The NP-completeness of SAT asserts that no algorithm for this problem can be efficient in the worst case, under the hypothesis P= NP. Nevertheless, in practice many efficient algorithms exist for solving the SAT problem.
Satisfiability problems have direct applications in various domains such as opera- tions research, artificial intelligence and system architecture. For example, in opera- tions research, the graph-coloring problem can be modeled as an instance of SAT.To decide whether a graph with n vertices can be colored with k colors, we consider k×n Boolean variables, xij , i =1,...,n, j =1,...,k,wherexij takes the value true if
Chapter written by Cristina BAZGAN. 4 Combinatorial Optimization 2 and only if the vertex i is assigned the color j.Hoos [HOO 98] studied the effective- ness of various modelings of the graph-coloring problem as a satisfiability problem where we apply a specific local search algorithm to the instance of the obtained sat- isfiability problem. The Steiner tree problem, widely studied in operations research, contributes to network design and routing applications. In [JIA 95], the authors re- duced this problem to a problem that consists of finding an assignment that maximizes the number of satisfied constraints. Certain scheduling problems have been solved by using modeling in terms of a satisfiability problem [CRA 94]. Testing various proper- ties of graphs or hypergraphs is also a problem that reduces to a satisfiability problem. In artificial intelligence, an interesting application is the planning problem that can be represented as a set of constraints such that every satisfying assignment corresponds to a valid plan (see [KAU 92] for such a modeling). Other applications in artificial intelligence are: learning from examples, establishing the coherence of a system of rules of a knowledge base, and constructing inferences in a knowledge base. In the design of electrical circuits, we generally wish to construct a circuit with certain func- tionalities (described by a Boolean function) that satisfy various constraints justified by technological considerations of reliability or availability, such as minimizing the number of gates used, minimizing the depth of the circuit or only using certain types of gates.
Satisfiability problems also have other applications in automatic reasoning, com- puter vision, databases, robotics, and computer-assisted design. Gu, Purdom, Franco and Wah wrote an overview article [GU 97] that cites many applications of satisfiabil- ity problems (about 250 references).
Faced with a satisfiability problem, we can either study it from the theoretical point of view (establish its exact or approximate complexity, construct algorithms that guarantee an exact or approximate solution), or solve it from the practical point of view. Among the most effective methods for the practical solution of satisfiability problems are local search, Tabu search, and simulated annealing. For further details, refer to [GU 97] and [GEN 99], which offer a summary of the majority of practical algorithms for satisfiability problems.
In this chapter, we present the principal results of exact and approximation com- plexity for satisfiability problems according to the type of Boolean functions that par- ticipate in the constraints. Our goal is not to present exhaustively all the results that exist in the literature but rather to identify the most studied problems and to introduce the basic concepts and algorithms. The majority of satisfiability problems are hard. It is therefore advantageous, both from the theoretical and practical points of view, to identify some specific cases that are easier. We have chosen to present the most stud- ied specific cases: planar instances, instances with a bounded number of occurrences of each variable, and dense instances. Several optimization problems can be modeled as a satisfiability problem with an additional global constraint on the set of feasible so- lutions. In particular, the MIN BISECTION problem, whose approximation complexity Optimal Satisfiability 5 has not yet been established, can be modeled as a satisfiability problem where the set of feasible solutions is the set of the balanced assignments (with as many variables set to 0 as to 1). We also present a few results obtained on satisfiability problems under this global constraint.
Readers who wish to acquire a deeper knowledge of the complexity of satisfiability problems should consult the monograph by Creignou, Khanna and Sudan [CRE 01], where the proofs of the majority of important results in this domain can be found and that cover, besides the results presented here, other aspects such as counting complex- ity and function representation complexity, as well as other satisfiability problems. Note also that there is an electronic compendium by Crescenzi and Kann [CRE 95b], which regroups known results of approximation complexity for optimization prob- lems, in particular for satisfiability problems.
This chapter is structured as follows. In section 1.2, we introduce the types of Boolean functions that we will use and we define the decision and optimization prob- lems considered. In section 1.3, we study decision problems, and in section 1.4, max- imization and minimization problems. We then discuss a few specific instances of sat- isfiability problems: planar instances (section 1.5.1), dense instances (section 1.5.2), and instances with a bounded number of occurrences of each variable (section 1.5.3). We also present the complexity of satisfiability problems when the set of feasible so- lutions is restricted to balanced assignments (section 1.6). We close our chapter with a brief conclusion (section 1.7).
1.2. Preliminaries
An instance of a satisfiability problem is a set of m constraints C1,...,Cmdefined on a set of n variables x1,...,xn.A constraint Cj is the application of a Boolean function f : {0, 1}k →{0,1}toasubset of variables x ,...,x ,wherei ,...,i ∈ i1 ik 1 k {1,...,n}.This constraint is also expressed as f(x ,...,x ). Anassignment x = i1 ik i v ,fori=1,...,n,wherev ∈{0,1},satisfies the constraint f(x ,...,x )if and i i i1 ik only if f(v ,...,v )=1. i1 ik
Aliteral is a variable xi (positive literal) or its negation x¯i (negative literal). EXAMPLE 1.1.– A few examples of Boolean functions used to define constraints: – T (x)=x, F (x)=¯x; k –ORi (x1,...,xk)=¯x1∨...∨x¯i∨xi+1 ∨ ...∨xk,wherei krepresents the number of negative literals in the disjunction; k – ANDi (x1,...,xk)=¯x1∧...∧x¯i ∧xi+1 ∧ ...∧xk,wherei krepresents the number of negative literals in the conjunction; k – XOR (x1,...,xk)=x1⊕... ⊕ xk,where⊕represents the “exclusive or” operation (0 ⊕ 0=0,1⊕0=1,0⊕1=1,1⊕1=0); 6 Combinatorial Optimization 2
k – XNOR (x1,...,xk)=x1⊕...⊕xk.
Aconstraint can also be represented as a Boolean expression that can be in various forms. An expression is in conjunctive normal form (CNF) if it is in the form c1 ∧
...∧cm,where each ct is a disjunctive clause, that is in the form t1 ∨...∨ tp,where
ti,i=1,...,pare literals. An expression is in disjunctive normal form (DNF) if it is in the form c1 ∨ ...∨cm,where each ct is a conjunctive clause, that is in the form
t1 ∧ ...∧ tp,where ti,i=1,...,pare literals. A kCNF (or kDNF) expression is a CNF (or DNF) expression in which each clause contains at most k literals.
Note that if each constraint of a satisfiability problem is represented by a CNF expression, the set of constraints of the problem can itself be represented by a CNF expression that corresponds to the conjunction of the previous expressions.
We consider various satisfiability problems according to the type of Boolean func- tions used to define the constraints. Let F be a finite set of Boolean functions. A F-set of constraints is a set of constraints that only use functions that belong to F.An assignment satisfies an F-set of constraints if and only if it satisfies each constraint in the constraint set.
1.2.1. Constraint satisfaction problems: decision and optimization versions
In this section we define the classes of problems that we are going to study. This concerns decision and optimization versions of satisfiability problems.
The decision version of a problem consists of establishing whether this problem allows at least one solution; its search version consists of finding a solution if any exist. The optimization version of a problem consists of finding a solution that maximizes or minimizes a suitable function.
DEFINITION 1.1.– The satisfiability problem SAT(F) consists of deciding whether there exists an assignment that satisfies an F-set of constraints. The search problem associated with the decision problem SAT(F) consists of finding an assignment that satisfies an F-set of constraints if such an assignment exists or then returning “no” otherwise.
In this chapter, we will see that whenever we can solve the decision problem SAT(F) in polynomial time, we can also find a solution for the satisfiable instances and therefore solve the associated search problem in polynomial time.
It is normal practice to distinguish certain variants of SAT(F) where each function of F depends on at most (or exactly) k variables. These variants are expressed as kSAT(F)(orEkSAT(F)). Optimal Satisfiability 7
We now present a few classical decision problems as well as the corresponding satisfiability problem SAT(F): –SAT is the problem that consists of deciding whether a set of disjunctive clauses defined on n Boolean variables is satisfiable. It corresponds to the SAT(F) problem, k where F is the set of ORi functions, for k n. –CONJ is the problem that consists of deciding whether a set of conjunctive clauses defined on n Boolean variables is satisfiable. It corresponds to the SAT(F) k problem, where F is the set of ANDi functions, for k n. –LIN2 is the problem that consists of deciding whether a set of linear equations defined on n Boolean variables is satisfiable. It corresponds to the SAT(F) problem, where F is the set of XORk, XNORk functions, for k n. –2SAT is the version of SAT where each disjunctive clause has at most two literals, k and it corresponds to 2SAT(F), where F is the set of ORi functions, for k 2. –E3SAT is the version of SAT where each disjunctive clause has exactly three 3 3 3 3 literals, and it corresponds to SAT({OR0, OR1, OR2, OR3}).
DEFINITION 1.2.– The maximization problem MAX SAT(F) consists of establish- ing an assignment that satisfies a maximum number of constraints from an F-set of constraints.
For example, the MAX CUT problem, which consists of partitioning the set of vertices of a graph into two parts such that the number of edges whose extremities belong to different parts is maximum, can be formulated as a problem of the type MAX SAT({XOR2}) as follows. Considering a graph G, an instance of MAX CUT,we associate with each vertex i avariablexiand with each edge (i, j) of G the constraint 2 XOR (xi,xj).
DEFINITION 1.3.– The minimization problem MIN SAT DELETION(F) consists of establishing an assignment that minimizes the number of non-satisfied constraints from an F-set of constraints, which corresponds with the minimum number of con- straints to remove so that the remaining constraints are satisfied.
MIN SAT DELETION(F) allows us to model certain minimization problems natu- rally. For example, the s-t MIN CUT problem in a non-directed graph, which consists of partitioning the set of vertices of a graph into two parts such that s and t belong to different parts and such that the number of edges whose extremities belong to differ- ent parts is minimum, can be formulated as a problem of the type MIN SAT DELE- TION({XNOR2}∪{T,F}) asfollows. Considering a graph G, an instance of s-t MIN CUT, we associate with each vertex i avariablexiand with each edge (i, j) of G the 2 constraint XNOR (xi,xj).Furthermore, we add the constraints T (xs) and F (xt). 8 Combinatorial Optimization 2
COMMENT 1.1.– 1) The problems MAX SAT(F)andMIN SAT DELETION(F) are clearly related. Indeed, considering an instance I of MAX SAT(F) with m constraints, an optimal solution for the instance I of MAX SAT(F)ofvalueoptMAX SAT(F)(I) is also an optimal solution of the instance I of the MIN SAT DELETION(F) problem of value optMIN SAT DELETION(F)(I)= m-optMAX SAT(F)(I).Therefore, the exact complexities of MAX SAT(F)andMIN SAT DELETION(F) coincide. However, the approximation complexities of the two problems can be very different as we will see in what follows. 2) In the literature, we also define the MIN SAT(F) problem that consists of es- tablishing an assignment that minimizes the number of satisfied constraints. For ex- ample, in the compendium of Crescenzi and Kann [CRE 95b], the MIN SAT prob- lem consists of establishing an assignment that minimizes the number of satisfied clauses from a set of disjunctive clauses. Note that MIN SAT(F) is equivalent, from the exact and approximation point of view, to MIN SAT DELETION(F ), where F is the set of functions that are complementary to the functions of F. For example, finding an assignment that minimizes the number of constraints satisfied among the constraints x1 ∨ x2,x1 ∨ x¯3,x2 ∨ x3,x¯1 ∨ x¯2 is equivalent to finding an assign- ment that minimizes the number of non-satisfied constraints among the constraints x¯1 ∧ x¯2, x¯1 ∧ x3, x¯2 ∧ x¯3,x1 ∧ x2.Thus the MIN SAT problem is equivalent to MIN SAT DELETION(F) where the constraints are conjunctive clauses (the problem called MIN CONJ DELETION). In what follows, we will consider only the MIN SAT DELETION(F) problem.
1.2.2. Constraint types
The complexity of SAT(F) as well as the exact and approximation complexities of MAX SAT(F)andMIN SAT DELETION(F) depend on the types of Boolean func- tions of the set F. In this section, we describe the types of Boolean functions that have been most studied and that we will discuss in the rest of the chapter.
A Boolean function f is: – 0-valid if f(0,...,0) = 1; – 1-valid if f(1,...,1) = 1; – Horn if it can be expressed as a CNF expression that has at most one positive literal in each clause; – anti-Horn if it can be expressed as a CNF expression that has at most one nega- tive literal in each clause; – affine if it can be expressed as a conjunction of linear equations on the Galois body GF (2), that is as a conjunction of equations of type xi1 ⊕ ...⊕xip =0or xj1 ⊕ ...⊕xjq =1; Optimal Satisfiability 9
– bijunctive if it can be expressed as a 2CNF expression;
– 2-monotone if it can be expressed as a DNF expression in the form xi1 ∧...∧xip or x¯ 1 ∧ ...∧x¯ q or (xi1 ∧ ...∧xip)∨(¯x 1 ∧ ...∧x¯ q).Note that a 2-monotone function is both Horn and anti-Horn; – complement-closed if for every assignment v we have f(v)=f(¯v),wherev¯is the complementary assignment to v.
We extend the previous notions to a set F of functions where all the functions of F have the required property. For example, if each function of F is Horn then the set F is Horn and the SAT(F) problem is called HORN SAT.Horn expressions play an important part in artificial intelligence for developing expert systems or formalizing knowledge bases. They also represent the base logic of Prolog.
The notation used in the literature for the SAT(F) problem when F is affine is LIN2. k-LIN2isthekSAT(F) problem where F is affine and Ek-LIN2isthevariant of k-LIN2 where each equation depends on exactly k variables. An instance of LIN2 is 0-homogenous (or 1-homogenous) if all its linear equations have their free terms equal to 0 (or 1 respectively).
MONOTONE-SAT and MONOTONE-kSAT are the variants of SAT and kSAT where every clause contains only positive literals or contains only negative literals.
We will consider other variants of SAT(F) in this chapter:
–TheNAE3SAT problem is SAT({f }), where f is of arity 3 and f(x1,x2,x3)= 1 if and only if the three variables do not have the same value; more exactly, f(0, 0, 0) = f(1, 1, 1) = 0,otherwiseftakes the value 1.
–The1IN3SAT problem is SAT({g}), where g is of arity 3 and g(x1,x2,x3)=1if and only if exactly one of the three variables has the value 1; more exactly, g(1, 0, 0) = g(0, 1, 0) = g(0, 0, 1) = 1,otherwisegtakes the value 0.
COMMENT 1.2.– For certain variants of the SAT(F) problem, the set of constraints can be represented in an equivalent way by an expression that puts these constraints into conjunction. In the associated optimization problems, such as MAX SAT(F) and MIN SAT DELETION(F), we use only the expression in the form of a set of constraints in order to be able to count the number of satisfied constraints.
We now present a few variants of optimization problems used in the rest of the chapter: –MAX SAT, given a set of disjunctive clauses defined on n Boolean variables, con- sists of finding an assignment that maximizes the number of satisfied clauses. MAX k SAT therefore corresponds to the MAX SAT(F) problem where F is the set of ORi functions, for k n. 10 Combinatorial Optimization 2
–MIN SAT DELETION, given a set of disjunctive clauses defined on n Boolean variables, consists of finding an assignment that minimizes the number of non-satisfied clauses. MIN SAT DELETION therefore corresponds to the MIN SAT DELETION(F) k problem where F is the set of ORi functions, for k n. –MAX CONJ, given a set of conjunctive clauses defined on n Boolean variables, consists of finding an assignment that maximizes the number of satisfied clauses. MAX CONJ therefore corresponds to the MAX SAT(F) problem where F is the set k of ANDi functions, for k n. –MIN CONJ DELETION, given a set of conjunctive clauses defined on n Boolean variables, consists of finding an assignment that minimizes the number of non-satisfied clauses. MIN CONJ DELETION therefore corresponds to the MIN SAT DELETION k (F) problem where F is the set of ANDi functions, for k n. –MAX LIN2, given a set of linear equations defined on n Boolean variables, con- sists of finding an assignment that maximizes the number of satisfied equations. MAX LIN2 therefore corresponds to the MAX SAT(F) problem where F is the set of XORk, XNORk functions, for k n. –MIN LIN2DELETION, given a set of linear equations defined on n Boolean variables, consists of finding an assignment that minimizes the number of non- satisfied equations. MIN LIN2DELETION therefore corresponds to the MIN SAT DELETION(F) problem where F is the set of XORk, XNORk functions, for k n; – The problems MAX kSAT,MAX EkSAT,MAX kCONJ,MAXEkCONJ,MAX k-LIN2, and MAX Ek-LIN2, as well as the corresponding MIN DELETION versions, are defined in a similar way on clauses or equations of size (at most) k.
1.3. Complexity of decision problems
In this section we study the complexity of SAT(F) decision problems according to the type of functions of the set F.
SAT was the first problem to be shown to be NP-complete by Cook [COO 71] and Levin [LEV 73]. We can easily reduce SAT to kSAT, k 3, which implies the NP-completeness of kSAT,fork 3.However,2SAT is polynomial [COO 71].
THEOREM 1.1.– 2SAT is solvable in polynomial time.
Proof. Let I be an instance of 2SAT with m clauses C1,...,Cm and n variables x1,...,xn. Wewillconstruct a directed graph GI with 2n vertices v1, v¯1,...,vn,v¯n, where vi (or v¯i respectively) corresponds to xi (or x¯i respectively). For a literal i ¯ (respectively i), let us express by wi (or w¯i respectively) the corresponding vertex. In this way, if i = xi then wi = vi and w¯i =¯vi,andif i =¯xithen wi =¯viand w¯i = vi.Each clause made up of one single literal is replaced by the equivalent clause ∨ . For each clause 1 ∨ 2, that is equivalent to the logical implications