polynomial time approximation schemes for euclidean tsp and other geometricproblemsnew

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1、PolynomialTimeApproximationSchemesforEuclideanTSPandotherGeometricProblemsSanjeevAroraPrincetonUniversitytheTSP'snastiness(cf.PAbstractDcompleteness[37]andPLS-completeness[20]).WepresentapolynomialtimeapproximationschemeforButTSPinstancesarisinginpracticeareusuallyquit

2、eEuclideanTSPin2<.Givenanynnodesintheplanespecial,sothehardnessresultsmaynotnecessarilyapplytoand>0,thescheme®ndsa(1+)-approximationthem.InmetricTSPthenodeslieinametricspace(i.e.,thetotheoptimumtravelingsalesmantourintimeO(1=).ndistancessatisfythetriangleinequality).

3、InEuclideanTSPd2Whenthenodesareind<,therunningtimeincreasesto(ormoregenerally,inthenodesliein<

4、metricTSP.approximationinpolynomialtime.Unfortunately,evenEuclideanTSPisNP-hard(Papadim-Wealsogivesimilarapproximationschemesforahostitriou[35],Garey,Graham,andJohnson[12]).ThereforeofotherEuclideanproblems,includingSteinerTree,k-TSP,algorithmdesignerswereleftwithnochoi

5、cebuttocon-Minimumdegree-kspanningtree,k-MST,etc.(Thislistmaysidermoremodestnotionsofaªgood"solution.Karp[24],getlonger;ourtechniquesarefairlygeneral.)Theprevi-inaseminalworkonprobabilisticanalysisofalgorithms,ousbestapproximationalgorithmsforalltheseproblemsshowedthatw

6、henthennodesarepickeduniformlyandachievedaconstant-factorapproximation.independentlyfromunitsquare,thenthe®xeddissectionAllouralgorithmsalsowork,withalmostnomodi®ca-heuristicwithhighprobability®ndstourswhosecostistion,whendistanceismeasuredusinganygeometricnormwithin1+

7、ofoptimal(where>0isarbitrarilysmall).(suchasChristo®des[9]designedanapproximationalgorithmthat`forp1orotherMinkowskinorms).poneveryinstanceofmetricTSPcomputesatourofcostatmost1:5timestheoptimum.1IntroductionTwodecadesofresearchfailedtoimproveuponChristo®des'algorithmf

8、ormetricTSP.Butsomere-searcherscontinuedtohopethatevenaPTASmightex-IntheTravelingSalesmanProblem(ªTSP"),weareg