2023年6月24日发(作者：)

CSCI280Fall2012NP-CompletenessMainStepsToproveaproblemXisNP-complete,youneedtoshststepistypicallydonebyshowingthatY≤1:hatthereisanefficientverifirwords,describeaverifiersuchthatforanyyesinstanceofX,thereexistsacertificatethattheverifierwillaccept,andforanynoinstanceofX,nosuchcertifiningtimeoftheverifier(andhencethesizeofthecertificate)lly,asolutiontothegivenproblemisasufficientcertifiepisverybrief,2:hatproblemYyouarereducingtoX;youneedtoshowthatY≤useanyproblemerydefinitionofNP-complete,ifXisNP-complete,r,tenusefultospendsometimethinkingaboutwhichoftheproblems3:atainstanceofYcanbesolvedusingapolynomialnumberofoperations,:ItisveryeasytogetmixedupandinsteadprovethatX≤unately,thisisnotwhatyouwanttoshow(wealreadyknowthatYisNP-complete).Typically,youwillshowhowtosolveYbyconstructingasingleinputtotheblackboxforX,andtypically,r,thisisnotalwaysthecase(inwhichcase,definethecorrespondencebetweentheoutputoftheblackboxforXandthedesiredsolutionforY.)Step4:waystrivialtocomeupwithanalgorithmthatsatisfisomethingthatsatisfi4a:Provethesolutionyoufiatifyouralgorithmreturns“yes,”4b:ProvethatyoufiatifyouaregivenayesinstanceofY,thenyouralgorithmreturns“yes”.Step5:rethatyououldallowyoutoconcludethatsinceyouralgorithmrunsinpolynomialtime,Y≤isNP-complete,andsincewealsohaveshownthatXisinNP,ple:INDEPENDENTSET(IS)AninputtoindependentsetisgivenbyG=(V,E)andanintegerk≥listodeterminewhethertnysetS,ourverifierwillcheckthatSindeedcontainsatleastkvertices(inO(k)time),andthatnotwoverticesofSareadjacent(inO(km)time).Forayesinstance(G,k),ourcertificateisanyindependentsetofsizekinG;ourverifiinstance(G,k),itisclearthatnosetofkverticesofGcanmakeourverifithat3SAT≤ninstanceof3SAT(aformulaϕ=C1∧···∧Cm,whereeachCiisthedisjunctionof3termsdrawnfromvariablesx1,x2,...,xnandtheirnegationsx1,x2,...,xn),weconstructaninstance(G,k)thesetoftermsinϕ;thatis,foreachtermtjineachclause,attherearemultiplenodeswiththelabelx