approximate strong separation with application in fractional graph coloring and preemptivenew

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1、Approximatestrongseparationwithappli ationinfra tionalgraph oloringandpreemptives hedulingylausansen
nstitutfur
nformatikundpraktis heathematik,Universitatzuiel,Germanykjinformatik.uni-kiel.deAbstra t
nthispaperweshowthatapproximationalgorithmsfortheweight

2、edindependentsetands-dimensionalknapsa kproblemwithratioa anbeturnedintoapproximationalgorithmswiththesameratioforfra tionalweightedgraph oloringandpreemptiveresour e onstraineds heduling.
nordertoobtaintheseresults,wegeneralizeknownresultsbyGrots hel,ovaszandS

3、 hrijveronseparation,non-emptinesstest,optimizationandviolationinthedire tionofapproximability.1
ntrodu tionThefra tionalgraph oloringisdenedasfollows(seealso[10,15,24,26,27,28,29℄).etG=(V;E)beagraphwithapositiveweightwforea hvertexv2V.et
bethesetofvallindepe

4、ndentsetsofG.Theweightedfra tional oloringproblem onsistsofassigninganon-negativerealvaluextoea hindependentset
ofGsu hthatea hvertexv2Vis ompletely
overedbyindependentsets ontainingv(i.e.thesumoftheirvaluesisatleastw)andthevtotalvaluexisminimized.Grots heleta

5、l.[15℄provedthattheweightedfra tional oloring

problemis-hardforgeneralgraphs,but anbesolvedinpolynomialtimeforperfe tgraphs.Theyhaveprovedthefollowinginterestingresult:Foranygraph lassG,iftheproblemof omputing(G;w)(theweightofthelargestweightedindependentseti

6、nG)forgraphsG2Gis-hard,thentheproblemofdeterminingtheweightedfra tionalhromati number(G;w)fisalso-hard.Thisgivesanegativeresultoftheweightedfra tional oloringproblemevenforplanar ubi graphsandunitdiskgraphs.Furthermore,iftheweightedindependentsetproblemforg

7、raphsinGispolynomial-timesolvable,thentheweightedfra tional oloringproblemforG analsobesolvedinpolynomialtime.Therstinapproximabilityresultfortheunweightedversionoftheproblem(i.e.wherew=1vforea hvertexv2V)wasobtainedbyundandYannakakis[24℄whoprovedthatthereexis

8、tsaÆ>0su hthatthereisnopolynomial-timeapproximationalgorithmfortheproblemwithÆapproximationration,unless=.Feigeandilian[10℄showedthatthefra tionalhromati 1numbe