Generalized Network Improvement & Packing Problems PDF
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Michael Holzhauser Generalized Network Improvement and Packing Problems Michael Holzhauser Technische Universität Kaiserslautern Germany Vom Fachb ereich Mathematik der Technischen Universität Kaiserslautern zur Verleihung des akademischen Grades Doktor der Naturwissenschaften (Doctor rerum naturalium, Dr. rer. nat.) genehmigte Dissertation, 2016 D 386 Erstgutachter: Prof. Dr. Sven O. Krumke Zweitgutachter: Prof. Dr. Andreas Bley Tag der Disputation: 19. August 2016 ISBN 978-3-658-16811-7 ISBN 978-3-658-16812-4 (eBook) DOI 10.1007/978-3-658-16812-4 Library of Congress Control Number: 2016961699 Springer Spektrum © Springer Fachmedien Wiesbaden GmbH 2016 This work is subject to copyright. 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Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer Spektrum imprint is published by Springer Nature The registered company is Springer Fachmedien Wiesbaden GmbH The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany Abstract Networkflowproblemsandpackingproblemsingeneralaretwoofthemostinvesti- gatedclassesofproblemsindiscreteoptimization.Inthelastdecades,manycombina- torialalgorithmshavebeendevelopedandsteadilyimprovedthatcomputeexactor approximatesolutionsfortheseproblems.Althoughthesealgorithmsallowtosolve manyrealworldapplications,mostofthemarehighlytailoredtotheinherentstruc- tureoftheunderlyingproblemsanddonotadmitevenslightvariationsorextensions tothese. Inthisthesis,weinvestigatesuchextensionsandvariationsofknownnetworkflow and packing problems with respect to their complexity and approximability. In the budget-constrainedminimumcostflowproblem,oneseekstodetermineaminimumcost flowsubjecttoabudgetconstraintbasedonasecondkindofcosts.Forthisproblem, we study efficient exact and approximate combinatorial algorithms. We also investi- gatetwodiscretevariants,whichcanbeinterpretedasnetworkimprovementproblems in which the edge capacities in the underlying network are allowed to be modified. Althoughtheproblembecomeshardtosolveinthesediscretesettings,weareableto deriveexactandapproximatealgorithmsbyexploitinganinterestingconnectiontoa novelvariantofthetraditionalknapsackproblem. Wealsoinvestigatetwoextensionsofthetraditionalmaximumflowproblem.Inthe maximumflowproblemingeneralizedprocessingnetworks,theaimistodetermineamax- imum flow in which the flow on each edge is additionally bounded by a dynamic capacity that depends on the total amount of flow leaving the starting node of the edge.Althoughthisproblemisashardtosolveasanylinearfractionalpackingprob- lem,weareabletoadaptalgorithmsforthetraditionalmaximumandminimumcost flow problem. Finally, we investigate an extension of the traditional maximum flow probleminwhichtheflowleavinganedgeisdescribedbyaconvexincreasingfunc- tion of the flow entering the edge. While the problem becomes hard to solve and approximateeveninitsmostsimpleform,weareabletoderiveexactalgorithms. Beyond these problems, we investigate the connection between network flow prob- lems and packing problems in general by extending a well-known framework for derivingefficientapproximationalgorithmsforpackingproblemstoalargeclassof networkflowproblems. Contents 1 Introduction 1 2 Preliminaries 5 2.1 BasicNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 TheoryofComputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 GraphTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 NetworkFlowProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 ApproximationAlgorithms. . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 FractionalPackingandParametricSearchFrameworks 19 3.1 ParametricSearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 FractionalPackingFramework. . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 AGeneralizedFramework . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3.1 MinimizingOracles. . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3.2 SignOracles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.3 SeparationOracles . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4 Budget-ConstrainedMinimumCostFlows: TheContinuousCase 39 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1.1 PreviousWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1.2 ChapterOutline. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3 NetworkSimplexAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.3.1 NotationandDefinitions . . . . . . . . . . . . . . . . . . . . . . 46 4.3.2 NetworkSimplexPivots . . . . . . . . . . . . . . . . . . . . . . . 48 4.3.3 OptimalityConditions . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3.4 TerminationandRunningTime. . . . . . . . . . . . . . . . . . . 53 4.4 BicriteriaInterpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4.1 AWeaklyPolynomial-TimeAlgorithm . . . . . . . . . . . . . . 62 4.4.2 AStronglyPolynomial-TimeAlgorithm . . . . . . . . . . . . . . 66 4.5 Approximability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5.1 GeneralGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5.2 AcyclicGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 XII Contents 5 Budget-ConstrainedMinimumCostFlows: TheDiscreteCase 77 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.1.1 PreviousWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.1.2 ChapterOutline. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3 IntegralCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3.2 Approximability . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.3.3 ExactAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.4 TheBoundedKnapsackProblemwithLaminarCardinalityConstraints 89 5.4.1 APolynomial-TimeApproximationScheme . . . . . . . . . . . 92 5.4.2 FullyPolynomial-TimeApproximationSchemes. . . . . . . . . 98 5.4.3 APolynomial-TimeSolvableSpecialCase. . . . . . . . . . . . . 107 5.5 BinaryCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6 GeneralizedProcessingNetworks 117 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.1.1 PreviousWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.1.2 ChapterOutline. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.3 StructuralResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.4 ComplexityandApproximability . . . . . . . . . . . . . . . . . . . . . . 126 6.4.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.4.2 Approximability . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.5 Series-ParallelGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.5.1 AugmentingonFlowDistributionSchemes . . . . . . . . . . . 134 6.5.2 AFasterApproach . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.6 IntegralFlows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7 ConvexGeneralizedFlows 155 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.1.1 PreviousWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.1.2 ChapterOutline. . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.3 StructuralResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.4 ComplexityandApproximability . . . . . . . . . . . . . . . . . . . . . . 167 7.4.1 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Contents XIII 7.4.2 Approximability . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.5 ExactAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.5.1 GeneralGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.5.2 AcyclicGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.5.3 Series-ParallelGraphs . . . . . . . . . . . . . . . . . . . . . . . . 181 7.5.4 Extension-ParallelGraphs . . . . . . . . . . . . . . . . . . . . . . 187 7.5.5 RestrictedExtension-ParallelGraphs. . . . . . . . . . . . . . . . 188 7.6 IntegralFlows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 8 Conclusion 193 Bibliography 195 Glossary 203 Index 207 List of Figures Figure2.1 Aseries-parallelgraphandapossibledecompositiontree. . . . 11 Figure3.1 Illustrationofthetwocasesthatmightoccurduringthesimu- lationoftheseparationoracle . . . . . . . . . . . . . . . . . . . . 34 Figure4.1 ThesituationifthetwocyclesC(e)andC(e)arenotedge-disjoint. 52 Figure4.2 ThethreepossiblecasesforthesetE0ofblockingedges . . . . . 56 Figure4.3 AcycleC(e)thatisinducedbytheenteringedgee2L. . . . . . 57 Figure4.4 ThecasethatE0(cid:18)P0=C(e)\C(e). . . . . . . . . . . . . . . . . 58 Figure4.5 TheobjectivespaceoftheinterpretationofBCMCFPR asabi- criteriaminimumcostflowproblem. . . . . . . . . . . . . . . . . 63 Figure4.6 Theprocedureforcomparingacandidatevaluefor(cid:21)totheset ofoptimalvalues(cid:3)(cid:3). . . . . . . . . . . . . . . . . . . . . . . . . . 65 Figure5.1 TheconstructedinstanceforthereductionofEvenOddPartition toBCMCFPN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Figure5.2 AninstanceofBCMCFPNandthecorrespondinginstanceofLKP 91 Figure5.3 Representationofalaminarfamilyofsetsasabinarytree. . . . 92 Figure5.4 The point x that is assumed to be too far away from the "- approximateparetofrontier. . . . . . . . . . . . . . . . . . . . . . 105 Figure5.5 ThenetworkforagiveninstanceofExactCoverBy3Sets . . . . 110 Figure6.1 AninstanceofMFGPNbasedonapackingLP. . . . . . . . . . . 128 Figure6.2 ThenetworkforagiveninstanceofExactCoverBy3Sets.. . . . 149 Figure6.3 ThenetworkforagiveninstanceofSubsetSum. . . . . . . . . . 151 Figure7.1 Anexampleofirrationalinflowsandoutflows . . . . . . . . . . 159 Figure7.2 ExampleapplicationofLemma7.7. . . . . . . . . . . . . . . . . . 162 Figure7.3 Asampleflowdecomposition.. . . . . . . . . . . . . . . . . . . . 166 Figure7.4 AsampleapplicationofLemma7.15.. . . . . . . . . . . . . . . . 168 Figure7.5 ThenetworkforagiveninstanceofExactCoverBy3Sets.. . . . 169 Figure7.6 ThenetworkforagiveninstanceofSubsetSum. . . . . . . . . . 172 Figure7.7 ThenetworkforagiveninstanceofSubsetSum. . . . . . . . . . 174 XVI ListofTables Figure7.8 Replacementofmaximumdirectedsubpathsonacyclebysin- gleedges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Figure7.9 Iterativecontractionofseriesandparalleltreesinthealgorithm. 184 Figure7.10 Aseriestreewithk=3leavesandtwoparalleltreeswithk1=4 andk3=3leaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 List of Tables Table3.1 Thesummarizedresultsforthegeneralizedpackingframework inChapter3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Table4.1 Thesummarizedresultsforthecontinuousbudget-constrained minimumcostflowprobleminChapter4 . . . . . . . . . . . . . 76 Table5.1 Thesummarizedresultsfortheintegralbudget-constrainedmin- imumcostflowprobleminChapter5 . . . . . . . . . . . . . . . 114 Table5.2 Thesummarizedresultsforthebinarybudget-constrainedmin- imumcostflowproblemChapter5 . . . . . . . . . . . . . . . . . 115 Table6.1 Thesummarizedresultsforthemaximumflowproblemingen- eralizedprocessingnetworksinChapter6 . . . . . . . . . . . . . 152 Table6.2 The summarized results for the integral maximum flow prob- lemingeneralizedprocessingnetworksinChapter6 . . . . . . 152 Table7.1 Thesummarizedresultsfortheconvexgeneralizedmaximum flowprobleminChapter7 . . . . . . . . . . . . . . . . . . . . . . 191 1 Introduction Network Flow and Packing Problems Networksareubiquitousinoureverydaylives—beittransportation,communication, orsocialnetworks.Ineachofthesenetworks,severalnodes(crossings,peers,persons) areconnectedtoeachotherviaedges(streets,cables,friendship).Inmostcases,oneis interestedinsomesolutionformovingentities(commodities,messages,information) from one place to another in this network. This solution — which we refer to as a networkflow—isrestrictedbyseveralconstraintslikelowerboundsontheamountof entitiesthatneedtobemovedorupperboundsontheamountofentitiesthatcanbe movedviaonesingleedge. Ontheotherhand,thequalityofasolutionismeasured bysomeobjectivefunction. Usually,oneisinterestedinsolutionsthattransportflow fromonepointinthenetworktoanotherinasomewhatmostcost-effectiveway. We refertotheproblemoffindingsuchasolutionastheminimumcostflowproblem.Stated asalinearprogram,thisproblemisgivenasfollows: X min ce(cid:1)xe (1.1a) e2XE X s.t. xe- xe=-bv forallv2V, (1.1b) e2(cid:14)-(v) e2(cid:14)+(v) 06xe6ue foralle2E. (1.1c) Thereby, xe denotes the amount of flow that is routed via edge e while ce and ue denotethecostandcapacityofthecorrespondingedges, respectively. Thevaluebv denotesthesupplyofentitiesateachnodev. Wegivemoreinsightsinthisgeneral formulationinthesubsequentchapter. Anotherpopularsetofproblemsistheclassofpackingproblems.Onefamousrepresen- tativeforthiskindofproblemistheknapsackproblem,inwhichoneisinterestedinthe bestwayofpackingitemswithaspecificprofitandweightintoaknapsackwithout exceedingamaximumweight. Initsmostgeneralform,forsomematrixA2Rm>0(cid:2)n withnon-negativeentriesandtwovectorsc2Rn andb2Rm withpositiveentries, >0 >0 a(fractional)packingproblemcanbestatedasfollows: maxcTx (1.2a) s.t.Ax6b, (1.2b) x>0. (1.2c) © Springer Fachmedien Wiesbaden GmbH 2016 M. Holzhauser, Generalized Network Improvement and Packing Problems, DOI 10.1007/978-3-658-16812-4_1 2 Introduction The knapsack problem complies with this form with the additional restriction that the variables are required to be integral. At a first glance, the minimum cost flow problem and a packing problem do not seem to have much in common. However, accordingtothewell-knownflowdecompositiontheorem, eachminimumcostflow canbedecomposedintoflowsonsimplepathsandcycles(cf. (Ahujaetal.,1993)). In otherwords,wecanseeeachminimumcostflowasa“bundle”of“packed”flowson pathsandcyclesthatdonotviolatetheedgecapacities.Hence,eachflowproblemcan beseenasapackingprobleminitscore,sobothproblemsareinfactstronglyrelated toeachother. Thiscentralobservationwillbeunderlinedandexploitedthroughout thisthesis. Contributions Theminimumcostflowproblemisoneofmostinvestigatedproblemsinthefieldof discreteoptimization. Alargevarietyofcombinatorialalgorithmshaveemergedover the last decades that make the problem tractable both from a theoretical as well as a practical point of view. However, since these algorithms became more and more tailoredtotheinherentstructureoftheminimumcostflowproblem, slightchanges tothisstructureoftenmaketheusageofthecorrespondingalgorithmsoreventheir underlyingideasimpossibletoapply. Anextensionofthemodelbynewconstraints oramodificationofthegivenconstraintsin(1.1)mayinfluencethecomplexityand approximability of the problem significantly. Similarly, although dynamic program- ming schemes and approximation algorithms are known for the knapsack problem, additionalconstraintstypicallyleadtomuchmoredifficultvariantsoftheproblem. Inthisthesis,weinvestigatenovelextensionstowell-knownnetworkflowandpack- ing problems. In particular, we are interested in results about the complexity and approximabilityoftheseproblemsandseektofindefficientcombinatorialalgorithms that exploit the underlying structure of the corresponding models, in contrast to highly generic simplex-type methods. Among others, this thesis addresses the fol- lowingmajorissues: (cid:15) Weinvestigateanetworkimprovementproblem,inwhichthecapacitiesoftheedges canbeupgradeduptoaspecificamountbyspendingaseparateupgradebudget. Thisproblemisequivalenttotheadditionofanadditionalbudgetconstraintof P theform e2Ebe(cid:1)xe6Btotheformulation(1.1). (cid:15) We address an extended network flow problem in which the amount of flow enteringanedgeisnotonlyboundedbythecapacityconstraints(1.1c),butmust
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