Generalized Network Improvement & Packing Problems

216 Pages·2016·1.96 MB·english

by  Michael Holzhauser

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
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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