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1、1Chapter 8NP and ComputationalIntractabilitySlides by Kevin Wayne.Copyright 2005 Pearson-Addison Wesley.All rights reserved.2Algorithm Design Patterns and Anti-PatternsAlgorithm design patterns.Ex.nGreed.O(n log n)interval scheduling.nDivide-and-conquer.O(n log n)FFT.nDynamic programming.O(n2)edit d
2、istance.nReductions.Algorithm design anti-patterns.nNP-completeness.O(nk)algorithm unlikely.nPSPACE-completeness.O(nk)certification algorithm unlikely.nUndecidability.No algorithm possible.8.1 Polynomial-Time Reductions4Classify Problems According to Computational RequirementsQ.Which problems will w
3、e be able to solve in practice?A working definition.Cobham 1964,Edmonds 1965,Rabin 1966 Those with polynomial-time algorithms.YesProbably noShortest pathLongest pathMin cutMax cut2-SAT3-SATMatching3D-matchingPrimality testingFactoringPlanar 4-colorPlanar 3-colorBipartite vertex coverVertex cover5Cla
4、ssify ProblemsDesiderata.Classify problems according to those that can be solved in polynomial-time and those that cannot.Huge number of fundamental problems have defied classification for decades.This chapter.Show that these fundamental problems are computationally equivalent.6Polynomial-Time Reduc
5、tionDesiderata.Suppose we could solve X in polynomial-time.What else could we solve in polynomial time?Reduction.Problem X polynomial reduces to problem Y if arbitrary instances of problem X can be solved using:nPolynomial number of standard computational steps,plusnPolynomial number of calls to ora
6、cle that solves problem Y.Notation.X P Y.Remarks.nWe pay for time to write down instances sent to black box instances of Y must be of polynomial size.nNote:Cook reducibility.7Polynomial-Time ReductionPurpose.Classify problems according to relative difficulty.Design algorithms.If X P Y and Y can be s
7、olved in polynomial-time,then X can also be solved in polynomial time.Establish intractability.If X P Y and X cannot be solved in polynomial-time,then Y cannot be solved in polynomial time.Establish equivalence.If X P Y and Y P X,we use notation X P Y.Reduction By Simple EquivalenceBasic reduction s
8、trategies.Reduction by simple equivalence.Reduction from special case to general case.Reduction by encoding with gadgets.9Independent SetINDEPENDENT SET:Given a graph G=(V,E)and an integer k,is there a subset of vertices S V such that|S|k,and for each edge at most one of its endpoints is in S?Ex.Is
9、there an independent set of size 6?Yes.Ex.Is there an independent set of size 7?No.independent set10Vertex CoverVERTEX COVER:Given a graph G=(V,E)and an integer k,is there a subset of vertices S V such that|S|k,and for each edge,at least one of its endpoints is in S?Ex.Is there a vertex cover of siz
10、e 4?Yes.Ex.Is there a vertex cover of size 3?No.vertex cover11Vertex Cover and Independent SetClaim.VERTEX-COVER P INDEPENDENT-SET.Pf.We show S is an independent set iff V S is a vertex cover.vertex coverindependent set12Vertex Cover and Independent SetClaim.VERTEX-COVER P INDEPENDENT-SET.Pf.We show
11、 S is an independent set iff V S is a vertex cover.nLet S be any independent set.nConsider an arbitrary edge(u,v).nS independent u S or v S u V S or v V S.nThus,V S covers(u,v).nLet V S be any vertex cover.nConsider two nodes u S and v S.nObserve that(u,v)E since V S is a vertex cover.nThus,no two n
12、odes in S are joined by an edge S independent set.Reduction from Special Case to General CaseBasic reduction strategies.Reduction by simple equivalence.Reduction from special case to general case.Reduction by encoding with gadgets.14Set CoverSET COVER:Given a set U of elements,a collection S1,S2,.,S
13、m of subsets of U,and an integer k,does there exist a collection of k of these sets whose union is equal to U?Ex:U=1,2,3,4,5,6,7 k=2S1=3,7S4=2,4S2=3,4,5,6S5=5S3=1S6=1,2,6,715SET COVERU=1,2,3,4,5,6,7 k=2Sa=3,7Sb=2,4Sc=3,4,5,6Sd=5Se=1Sf=1,2,6,7Vertex Cover Reduces to Set CoverClaim.VERTEX-COVER P SET-
14、COVER.Pf.Given a VERTEX-COVER instance G=(V,E),k,we construct a set cover instance whose size equals the size of the vertex cover instance.Construction.nCreate SET-COVER instance:k=k,U=E,Sv=e E:e incident to v nSet-cover of size k iff vertex cover of size k.adbefcVERTEX COVERk=2e1 e2 e3 e5 e4 e6 e7
15、8.2 Reductions via GadgetsBasic reduction strategies.Reduction by simple equivalence.Reduction from special case to general case.Reduction via gadgets.17Ex:Yes:x1=true,x2=true x3=false.Literal:A Boolean variable or its negation.Clause:A disjunction of literals.Conjunctive normal form:A propositional
16、formula that is the conjunction of clauses.SAT:Given CNF formula,does it have a satisfying truth assignment?3-SAT:SAT where each clause contains exactly 3 literals.Satisfiability183 Satisfiability Reduces to Independent SetClaim.3-SAT P INDEPENDENT-SET.Pf.Given an instance of 3-SAT,we construct an i
17、nstance(G,k)of INDEPENDENT-SET that has an independent set of size k iff is satisfiable.Construction.nG contains 3 vertices for each clause,one for each literal.nConnect 3 literals in a clause in a triangle.nConnect literal to each of its negations.k=3G193 Satisfiability Reduces to Independent SetCl
18、aim.G contains independent set of size k=|iff is satisfiable.Pf.Let S be independent set of size k.nS must contain exactly one vertex in each triangle.nSet these literals to true.nTruth assignment is consistent and all clauses are satisfied.Pf Given satisfying assignment,select one true literal from
19、 each triangle.This is an independent set of size k.k=3G20ReviewBasic reduction strategies.nSimple equivalence:INDEPENDENT-SET P VERTEX-COVER.nSpecial case to general case:VERTEX-COVER P SET-COVER.nEncoding with gadgets:3-SAT P INDEPENDENT-SET.Transitivity.If X P Y and Y P Z,then X P Z.Pf idea.Compo
20、se the two algorithms.Ex:3-SAT P INDEPENDENT-SET P VERTEX-COVER P SET-COVER.21Self-ReducibilityDecision problem.Does there exist a vertex cover of size k?Search problem.Find vertex cover of minimum cardinality.Self-reducibility.Search problem P decision version.nApplies to all(NP-complete)problems i
21、n this chapter.nJustifies our focus on decision problems.Ex:to find min cardinality vertex cover.n(Binary)search for cardinality k*of min vertex cover.nFind a vertex v such that G v has a vertex cover of size k*-1.any vertex in any min vertex cover will have this propertynInclude v in the vertex cover.nRecursively find a min vertex cover in G v.