180thisiscalledPolynomialtimeApproximationSchemePTASegcertainschedulingproblems 1intimethatispolynomialinn1 thisiscalledFullyPolynomialtimeapproximationSchemeFPTASegKnapsa ID: 212387
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CS105:Algorithms(Grad)ApproximationAlgorithms(continued)Feb21,2005 =1+",8"�0,thisiscalledPolynomial-timeApproximationScheme(PTAS),e.g.certainschedulingproblems. =1+"intimethatispolynomialin(n;1 "),thisiscalledFullyPolynomial-timeapproximationScheme(FPTAS),e.g.Knapsack,SubsetSum.Now,letusconsideranapproximationalgorithmforNP-Hardproblem,VertexCover.1.2ApproximationAlgorithmforVertexCoverGivenaG=(V;E),ndaminimumsubsetCV,suchthatC\covers"alledgesinE,i.e.,everyedge2EisincidenttoatleastonevertexinC: Figure1:AninstanceofVertexCoverproblem.Anoptimalvertexcoverisfb,c,e,i,gg. Algorithm1:Approx-Vertex-Cover(G) C ;1whileE6=;2pickanyfu;vg2EC C[fu;vgdeleteallegesincidenttoeitheruorvreturnC Asitturnsout,thisisthebestapproximationalgorithmknownforvertexcover.Itisanopenproblemtoeitherdobetterorprovethatthisisalowerbound.Observation:Thesetofedgespickedbythisalgorithmisamatching,no2edgestoucheachother(edgesdisjoint).Infact,itisamaximalmatching.Wecanthenhavethefollowingalternativedescriptionofthealgorithmasfollows.FindamaximalmatchingMReturnthesetofend-pointsofalledges2M. Page2of7 CS105:Algorithms(Grad)ApproximationAlgorithms(continued)Feb21,2005 1.3AnalysisofApproximationAlgorithmforVCClaim1:ThisalgorithmgivesavertexcoverProof:Everyedge2Misclearlycovered.Ifanedge,e=2Misnotcovered,thenM[fegisamatching,whichcontradicttomaximalityofM.Claim2:Thisvertexcoverhassize2minimumsize(optimalsolution)Proof: Figure2:AnotherinstanceofVertexCoveranditsoptimalcovershowninbluesquaresTheoptimumvertexcovermustcovereveryedgeinM.So,itmustincludeatleastoneoftheendpointsofeachedge2M,whereno2edgesinMshareanendpoint.Hence,optimumvertexcovermusthavesizeOPT(I)jMjButthealgorithmAreturnavertexcoverofsize2jMj,so8IwehaveA(I)=2jMj2OPT(I)implyingthatAisa2-approximationalgorithm.Weknowthattheoptimalsolutionisintractable(otherwisewecanprobablycomeupwithanalgorithmtondit).Thus,wecannotmakeadirectcomparisonbetweenalgorithmA'ssolutionandtheoptimalsolution.ButwecanproveClaim2bymakingindirectcomparisonsofA'ssolutionandtheoptimalsolutionwiththesizeofthemaximalmatching,jMj.WeoftenusethistechniqueforapproximationproofsforNP-Hardproblems,asyouwillseelateron.Butis =2atightboundforthisalgorithm?Isitpossiblethatthisalgorithmcandobetterthan2-approximation?Wecanshowthat2-approximationisatightboundbyatightexample:TightExample:Consideracompletebipartitegraphofnblacknodesononesideandnrednodesontheotherside,denotedKn;n.Noticethatsizeofanymaximalmatchingofthisgraphequalsn,jMj=n Page3of7 CS105:Algorithms(Grad)ApproximationAlgorithms(continued)Feb21,2005 Figure3:Kn;n-completebipartitegraphsotheApprox-Vertex-Cover(G)algorithmreturnsacoverofsize2n.A(Kn;n)=2nBut,clearlytheoptimalsolution=n.OPT(Kn;n)=nNotethatatightexampleneedstohavearbitrarilylargesizeinordertoprovetightnessofanalysis,otherwisewecanjustusebruteforceforsmallgraphsandAforlargeonestogetanalgorithmthatavoidthattightbound.Here,itshowsthatthisalgorithmgives2-approximationnomatterwhatsizenis.2ApproximationAlgorithms:TravelingSalesmanProblem2.1Lasttime: -approximationalgorithmsDenition:Foraminimization(ormaximization)problemP,Aisan -approximationalgorithmifforeveryinstanceIofP,A(I) OPT(I) (orOPT(I) A(I) ).Lasttimewesawa2-approximationforVertexCover[CLRS35.1].Todaywewillseea2-approximationfortheTravelingSalesmanProblem(TSP)[CLRS35.2].2.2DenitionAsalesmanwantstovisiteachofncitiesexactlyonceeach,minimizingtotaldistancetravelled,andreturningtothestartingpoint.TravelingSalesmanProblem(TSP).Input:acomplete,undirectedgraphG=(V;E),withedgeweights(costs)w:E�!R+,andwherejVj=n.Output:atour(cyclethatvisitsallnverticesexactlyonceeach,andreturningtostartingvertex)ofminimumcost. Page4of7 CS105:Algorithms(Grad)ApproximationAlgorithms(continued)Feb21,2005 2.3InapproximabilityResultforGeneralTSPTheorem:Foranyconstantk,itisNP-hardtoapproximateTSPtoafactorofk.Proof:RecallthatHamiltonianCycle(HC)isNP-complete(Sipser).ThedenitionofHCisasfollows.Input:anundirected(notnecessarilycomplete)graphG=(V;E).Output:YESifGhasaHamiltoniancycle(ortour,asdenedabove),NOotherwise.SupposeAisak-approximationalgorithmforTSP.WewilluseAtosolveHCinpolynomialtime,thusimplyingP=NP. Figure4:ExampleofconstructionofG0fromGforHC-to-TSP-approximationreduction.GiventheinputG=(V;E)toHC,wemodifyittoconstructthegraphG0=(V0;E0)andweightfunctionwasinputtoAasfollows(Figure4).LetalledgesofGhaveweight1.Completetheresultinggraph,lettingallnewedgeshaveweightLforsomelargeconstantL.ThealgorithmforHCisthen: Algorithm2:HC-Reduction(G) ConstructG0asdescribedabove.1ifA(G0)returnsa`small'costtour(kn)then2returnYES3ifA(G0)returnsa`large'costtour(L)then4returnNO5 ItthenremainstochooseourconstantLkn,toensurethatthe2casesareclearlydi erentiated.2.4ApproximationAlgorithmforMetricTSPDenition.Ametricspaceisapair(S;d),whereSisasetandd:S2�!R+isadistancefunctionthatsatises,forallu;v;w2S,thefollowingconditions.1.d(u;v)=02.d(u;v)=d(v;u) Page5of7 CS105:Algorithms(Grad)ApproximationAlgorithms(continued)Feb21,2005 3.d(u;v)+d(v;w)d(u;w)(triangleinequality)ForacompletegraphG=(V;E)withcostc:E�!R+,wesay\thecostsformametricspace"if(V;^c)isametricspace,where^c(u;v):=c(fu;vg).Giventhisrestriction(inparticular,theadditionofthetriangleinequalitycondition),wehavethefollowingsimpleapproximationalgorithmforTSP. Algorithm3:MetricTSPApprox(G) ComputeaweightedMSTofG.1RootMSTarbitrarilyandtraverseinpre-order:v1;v2;:::;vn.2Outputtour:v1!v2!!vn!v1.3 Figure5:ExampleMST,wheretheoutputtourwouldbe1!2!!17!1.2.5AnalysisofApproximationAlgorithmforMetricTSPOnaninstanceIofTSP,letuscompareA(I)toOPT(I),viatheintermediatevalueMST(I)(theweightoftheMST).Claim:ComparingA(I)toMST(I):A(I)2MST(I).Proof:LetbeafullwalkalongtheMSTinpre-order(thatis,werevisitverticesaswebacktrackthroughthem).InFigure5,wouldbethepathalongallthearrows,wrappingaroundtheentireMST,namely,1!2!1!3!4!5!4!6!4!!1.Itisclearthatcost()=2MST(I).Now,thetoutoutputbyAisasubsequenceofthefullwalk,sobythetriangleinequality:A(I)cost()=2MST(I)provingourclaim.Claim:ComparingOPT(I)toMST(I):OPT(I)MST(I).Proof:Letbeanoptimumtour,thatis,cost()=OPT(I).DeletinganedgefromresultsinaspanningtreeT,whosecostbydenitioniscost(T)MST(I).Hence,OPT(I)=cost()cost(T)MST(I) Page6of7 CS105:Algorithms(Grad)ApproximationAlgorithms(continued)Feb21,2005 asrequired.Combiningthese2claims,weget:A(I)2MST(I)2OPT(I)Hence,Aisa2-approximationalgorithmfor(Metric)TSP.2.6ConcludingRemarksItispossible(andrelativelyeasy)toimprovetheapproximationfactorto3/2forMetricTSP.Notethatintheoriginalwordingoftheproblem,withthesalesmantouringcities,thecost(distance)functionisinfactevenmorestructuredthanjustametric.Here,wehaveEuclideandistance,andasitturnsout,thisfurtherrestrictionallowsustogetaPTAS,althoughthisisamoredicultalgorithm. 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