Transcription of NP-complete problems
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Ndingshortestpathsandminimumspanningtree singraphs, matchingsinbipartitegraphs, maximumincreasingsub-sequences, maximum owsinnetworks, cient, becauseineach casetheirtimerequirementgrowsasa polynomialfunction(such asn,n2, orn3) ofthesizeof betterappreciatesuch ef cientalgorithms, considerthealternative:Inalltheseprob-le mswearesearchingfora solution(path,tree, matching, etc.)fromamonganexponentialpopulationofp ossibilities. Indeed,nboyscanbematchedwithngirlsinn!di fferentways, agraphwithnverticeshasnn 2spanningtrees, anda typicalgraphhasanexponentialnum-berof pathsfromstot. Alltheseproblemscouldinprinciplebesolved inexponentialtimebycheckingthroughallcan didatesolutions, onebyone. Butanalgorithmwhoserunningtimeis2n, orworse, is allbutuselessinpractice(seethenextbox).T hequestforef cientalgorithmsis about ndingcleverwaystobypassthisprocessofexha ustivesearch,usingcluesfromtheinputinord ertodramaticallynarrowdownthesearch thisquest,algorithmictech-niquesthatdefe atthespecterofexponentiality:greedyalgor ithms, dynamicprogramming,linearprogramming(whi ledivide-and-conquertypicallyyieldsfaste ralgorithmsforproblemswecanalreadysolvei npolynomialtime).
Chapter 8 NP-complete problems 8.1 Search problems Over the past seven chapters we have developed algorithms for nding shortest paths and minimum spanning trees in graphs, matchings in bipartite graphs, maximum increasing sub-sequences, maximum ows in networks, and so on. All these algorithms are efcient, because
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