P, NP, and NP-Completeness: The Basics of Computational Complexity

216 Pages·2010·0.59 MB·English

by  Oded Goldreich

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P,NP,andNP-Completeness
TheBasicsofComputationalComplexity
ThefocusofthisbookisthePversusNPQuestionandthetheoryofNP-completeness.
Italsoprovidesadequatepreliminariesregardingcomputationalproblemsandcompu-
tationalmodels.
ThePversusNPQuestionaskswhetherfindingsolutionsisharderthanchecking
thecorrectnessofsolutions.Analternativeformulationaskswhetherdiscoveringproofs
isharderthanverifyingtheircorrectness.Itiswidelybelievedthattheanswertothese
equivalent formulations is positive, and this is captured by saying that P is different
fromNP.
Although the P versus NP Question remains unresolved, the theory of NP-
completenessoffersevidencefortheintractabilityofspecificproblemsinNPbyshowing
thattheyareuniversalfortheentireclass.Amazinglyenough,NP-completeproblems
exist,andhundredsofnaturalcomputationalproblemsarisinginmanydifferentareas
ofmathematicsandscienceareNP-complete.
oded goldreich is a Professor of Computer Science at the Weizmann Institute of
ScienceandanIncumbentoftheMeyerW.WeisgalProfessorialChair.Heisaneditor
for the SIAM Journal on Computing, theJournal of Cryptology , and Computational
Complexity and previously authored the books Modern Cryptography, Probabilistic
Proofs and Pseudorandomness, the two-volume work Foundations of Cryptography,
andComputationalComplexity:AConceptualPerspective.

P, NP, and NP-Completeness
The Basics of Computational Complexity
ODED GOLDREICH
WeizmannInstituteofScience
CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
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Information on this title: www.cambridge.org/9780521192484
© Oded Goldreich 2010
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2010
ISBN-13 978-0-511-90793-7 eBook (EBL)
ISBN-13 978-0-521-19248-4 Hardback
ISBN-13 978-0-521-12254-2 Paperback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
toDana

Contents
ListofFigures pagexi
Preface xiii
Overview xvii
TotheTeacher xxi
NotationsandConventions xxv
MainDefinitionsandResults xxvii
1 ComputationalTasksandModels 1
TeachingNotes 2
1.1 Representation 3
1.2 ComputationalTasks 5
1.2.1 SearchProblems 5
1.2.2 DecisionProblems 6
1.2.3 PromiseProblems(anAdvancedComment) 8
1.3 UniformModels(Algorithms) 8
1.3.1 OverviewandGeneralPrinciples 9
1.3.2 AConcreteModel:TuringMachines 11
1.3.2.1 TheActualModel 12
1.3.2.2 TheChurch-TuringThesis 16
1.3.3 UncomputableFunctions 18
1.3.3.1 OntheExistenceofUncomputableFunctions 18
1.3.3.2 TheHaltingProblem 19
1.3.3.3 AFewMoreUndecidabilityResults 21
1.3.4 UniversalAlgorithms 22
1.3.4.1 TheExistenceofUniversalAlgorithms 23
1.3.4.2 ADetour:KolmogorovComplexity 24
1.3.5 Time(andSpace)Complexity 26
1.3.6 OracleMachinesandTuring-Reductions 29
vii
viii Contents
1.3.7 RestrictedModels 31
1.4 Non-UniformModels(CircuitsandAdvice) 31
1.4.1 BooleanCircuits 32
1.4.1.1 TheBasicModel 32
1.4.1.2 CircuitComplexity 35
1.4.2 MachinesThatTakeAdvice 36
1.4.3 RestrictedModels 37
1.4.3.1 BooleanFormulae 38
1.4.3.2 OtherRestrictedClassesofCircuits 39
1.5 ComplexityClasses 40
Exercises 41
2 ThePversusNPQuestion 48
TeachingNotes 49
2.1 EfficientComputation 50
2.2 TheSearchVersion:FindingversusChecking 53
2.2.1 TheClassPasaNaturalClassofSearchProblems 54
2.2.2 TheClassNPasAnotherNaturalClassofSearch
Problems 56
2.2.3 ThePversusNPQuestioninTermsofSearchProblems 57
2.3 TheDecisionVersion:ProvingversusVerifying 58
2.3.1 TheClassPasaNaturalClassofDecisionProblems 59
2.3.2 TheClassNPandNP-ProofSystems 59
2.3.3 ThePversusNPQuestioninTermsofDecisionProblems 62
2.4 EquivalenceoftheTwoFormulations 63
2.5 TechnicalCommentsRegardingNP 65
2.6 TheTraditionalDefinitionofNP 66
2.7 InSupportofPBeingDifferentfromNP 69
2.8 PhilosophicalMeditations 70
Exercises 71
3 Polynomial-timeReductions 74
TeachingNotes 75
3.1 TheGeneralNotionofaReduction 75
3.1.1 TheActualFormulation 76
3.1.2 SpecialCases 77
3.1.3 TerminologyandaBriefDiscussion 79
3.2 ReducingOptimizationProblemstoSearchProblems 81
3.3 Self-ReducibilityofSearchProblems 83