Approximation Techniques for Stochastic Combinatorial

135 Pages·2012·1.47 MB·English

Approximation Techniques for Stochastic
Combinatorial Optimization Problems
Ravishankar Krishnaswamy
CMU-CS-12-120
May 2012
SchoolofComputerScience
CarnegieMellonUniversity
Pittsburgh,PA15213
ThesisCommittee:
NikhilBansal
AvrimBlum
AnupamGupta,Chair
KirkPruhs
R.Ravi
Submittedinpartialfulfillmentoftherequirements
forthedegreeofDoctorofPhilosophy.
Copyright(cid:13)c 2012RavishankarKrishnaswamy
ThisresearchwassponsoredbytheNationalScienceFoundationundergrantnumbersCCF-0448095,CCF-0729022,
andCCF-0964474. Theviewsandconclusionscontainedinthisdocumentarethoseoftheauthorandshouldnotbe
interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S.
governmentoranyotherentity.
Keywords: Approximation Algorithms, Stochastic Optimization, Network Design, Schedul-
ing
Abstract
Thefocusofthisthesisisonthedesignandanalysisofalgorithmsforbasicprob-
lems in Stochastic Optimization, specifically a class of fundamental combinatorial
optimization problems where there is some form of uncertainty in the input. Since
many interesting optimization problems are computationally intractable (NP-Hard),
we resort to designing approximation algorithms which provably output good so-
lutions. However, a common assumption in traditional algorithms is that the exact
input is known in advance. What if this is not the case? What if there is uncertainty
intheinput?
With the growing size of input data and their typically distributed nature (e.g.,
cloud computing), it has become imperative for algorithms to handle varying forms
of input uncertainty. Current techniques, however, are not robust enough to deal
with many of these problems, thus necessitating the need for new algorithmic tools.
Answering such questions, and more generally identifying the tools for solving such
problems, is the focus of this thesis. The exact problems we study in this thesis are
the following: (a) the Survivable Network Design problem where the collection of
(source,sink) pairs is drawn randomly from a known distribution, (b) the Stochastic
Knapsack problem with random sizes/rewards for jobs, (c) the Multi-Armed Bandits
problem, where the individual Markov Chains make random transitions, and finally
(d) the Stochastic Orienteering problem, where the random tasks/jobs are located
at different vertices on a metric. We explore different techniques for solving these
problems and present algorithms for all the above problems with near-optimal ap-
proximation guarantees. We also believe that the techniques are fairly general and
havewiderapplicabilitythanthecontextinwhichtheyareusedinthisthesis.
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Acknowledgments
First and foremost, I would like to thank my advisor Anupam Gupta, for being
all that I sought in an advisor and more. He has been a constant source of encour-
agement and patience (needless to mention the technical ideas) throughout my PhD.
Iamdeeplyindebtedforallthetimeandefforthehasdevoted(oftengoingoutofhis
way)towardsmyresearchandgraduatelifehereatCarnegieMellon University.
I have also had the great fortune of having worked with some amazing collabo-
rators, from whom I have learned a lot. Many thanks to R. Ravi, Kirk Pruhs, Nikhil
Bansal, Viswanath Nagarajan, Sungjin Im, Benjamin Moseley, Kunal Talwar, Udi
Wieder, Aditya Bhaskara, Marco Molinaro, Maxim Sviridenko, Barna Saha, Danny
SegevandAmitKumar. I’malsogratefultoallmembersofthetheorygroupatCMU
forallgoodtimes—courses,theorylunches,andjustthepositivevibeingeneral.
The last five years have been truly memorable, and for that I am thankful to all
myfriends(andyoutoo,Divya). Lastbutnotleast,IwouldliketothankAmmaand
Appafortheirendlesssupport,encouragement,andprayersthroughoutthisjourney.
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Contents
1 Introduction 1
1.1 TraditionalApproximationAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 OnlineOptimizationanditsDrawbacks . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 StochasticOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 ModelI:Two-StochasticOptimizationwithRecourse . . . . . . . . . . . . . . . . 3
1.4.1 ModelDefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4.2 Example: StochasticSteinerTree . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 ModelII:Adaptive StochasticOptimization . . . . . . . . . . . . . . . . . . . . . 5
1.5.1 ModelDefinition,fortheStochasticKnapsackproblem . . . . . . . . . . . 6
1.5.2 ASimpleExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 ResultsinthisThesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6.1 Two-StageStochasticSurvivableNetworkDesign . . . . . . . . . . . . . 8
1.6.2 GeneralStochasticKnapsack . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6.3 Multi-ArmedBandits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6.4 StochasticOrienteering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 SomeNotesontheModelsStudied . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7.1 SpecifyingtheUncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7.2 ComputationalIntractability . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7.3 QualityofSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7.4 ComparisontoCompetitiveRatio . . . . . . . . . . . . . . . . . . . . . . 14
1.8 Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.9 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
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2 StochasticSurvivableNetworkDesign 17
2.1 OurResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 ChapterRoadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 OnlineSNDPonGeneralGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 OurTechniquesandtheirMotivation . . . . . . . . . . . . . . . . . . . . 21
2.4.3 EmbeddingintoBackbonedGraphs . . . . . . . . . . . . . . . . . . . . . 24
2.4.4 Online2-Edge-Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.5 ASmallCollection ofCoveringCycles . . . . . . . . . . . . . . . . . . . . 26
2.4.6 AugmentationusingHittingSets . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.7 OnlineAlgorithmusingHittingSets . . . . . . . . . . . . . . . . . . . . . 32
2.5 Two-StageStochasticSNDP OnGeneralGraphs . . . . . . . . . . . . . . . . . . 33
2.6 OnlineSNDPonMetricInstances . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7 StochasticSNDPonMetricInstances . . . . . . . . . . . . . . . . . . . . . . . . 37
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 StochasticKnapsack 43
3.1 OurResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 ChapterRoadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.1 BadnessDuetoCancellations . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.2 BadnessDuetoCorrelatedRewards . . . . . . . . . . . . . . . . . . . . . 47
3.4.3 IntuitionbehindourLP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 CorrelatedStochasticKnapsackwithoutCancellation . . . . . . . . . . . . . . . . 48
3.5.1 ProblemDefinitionsandNotation . . . . . . . . . . . . . . . . . . . . . . 48
3.5.2 LPRelaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5.3 RoundingAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 CorrelatedStochasticKnapsackwithCancellations . . . . . . . . . . . . . . . . . 52
3.6.1 RoundingAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.7 ProofsfromSection3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.7.1 MakingStocK-NoCancel FullyPolynomial . . . . . . . . . . . . . . . . . 55
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3.8 ProofsfromSection3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.8.1 StocKwithSmallSizes: AFullyPolytimeAlgorithm . . . . . . . . . . . . 57
3.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4 Multi-ArmedBandits 59
4.1 RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 ChapterRoadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.1 TheBenefitofPreemptioninNon-MartingaleBandits . . . . . . . . . . . 62
4.4 ApproximatingMAB onTree-like TransitionGraphs . . . . . . . . . . . . . . . . 64
4.4.1 GlobalTime-indexedLP . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4.2 TheRoundingAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.5 DetailsofPhaseI(fromSection4.4.2) . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.6 DetailsofPhaseII(fromSection4.4.2) . . . . . . . . . . . . . . . . . . . . . . . . 72
4.7 DetailsofPhaseIII(fromSection4.4.2) . . . . . . . . . . . . . . . . . . . . . . . . 74
4.8 MABswithArbitraryTransitionGraphs . . . . . . . . . . . . . . . . . . . . . . . 76
4.8.1 LayeredDAGscaptureallGraphs . . . . . . . . . . . . . . . . . . . . . . 76
4.8.2 OurTechniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.8.3 LPRelaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.8.4 ConvexDecomposition: TheAlteredPhaseI . . . . . . . . . . . . . . . . 77
4.8.5 PhasesIIandIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5 StochasticOrienteering 87
5.1 OurResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 ChapterRoadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3.1 OurApproach: ReductiontoKnapsackOrienteering . . . . . . . . . . . . 91
5.4 DefinitionsandNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5 The(Deterministic)KnapsackOrienteeringProblem . . . . . . . . . . . . . . . . 93
5.6 Non-AdaptiveStochasticOrienteering . . . . . . . . . . . . . . . . . . . . . . . . 94
5.7 BoundingtheAdaptivityGap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
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5.8 StochasticOrienteeringwithCancellations . . . . . . . . . . . . . . . . . . . . . . 103
5.9 StochasticOrienteeringwithCorrelatedRewards . . . . . . . . . . . . . . . . . . 104
5.9.1 TheNon-AdaptiveAlgorithmforCorrOrient . . . . . . . . . . . . . . . . 105
5.9.2 EvidenceofhardnessforCorrOrient . . . . . . . . . . . . . . . . . . . . . 110
5.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6 Conclusion 113
6.1 StochasticKnapsackwithCorrelatedJobs . . . . . . . . . . . . . . . . . . . . . . 113
6.2 SNDPwithVertexConnectivityRequirements . . . . . . . . . . . . . . . . . . . 114
6.3 StochasticRoutingProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Bibliography 117
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