Kernels for Feedback Arc Set In Tournaments

Stephane Bessy

Description

Kernels for Feedback Arc Set In Tournaments Stephane Bessy? Fedor V. Fomin† Serge Gaspers‡ Christophe Paul? Anthony Perez? Saket Saurabh† Stephan Thomasse? Abstract A tournament T = (V,A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T ? on O(k) vertices. In fact, given any fixed > 0, the kernelized instance has at most (2 + )k vertices. Our result improves the previous known bound of O(k2) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k- FAST. 1 Introduction Given a directed graph G = (V,A) on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic directed graph.

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Sujets

• Parameterized complexity
• Time complexity

Informations

• Publié par: pefav
• Nombre de lectures: 12
• Langue: English

Extrait

KernelsforFeedbackArcSetInTournaments Ste´phaneBessy ∗ FedorV.Fomin † SergeGaspers ‡ ChristophePaul ∗ AnthonyPerez ∗ SaketSaurabh † Ste´phanThomasse´ ∗ Abstract AtournamentT=(V,A)isadirectedgraphinwhichthereisexactlyonearc betweeneverypairofdistinctvertices.Givenadigraphonnverticesandaninteger parameterk,the FeedbackArcSet problemaskswhetherthegivendigraphhas asetofkarcswhoseremovalresultsinanacyclicdigraph.The FeedbackArc Set problemrestrictedtotournamentsisknownasthek -FeedbackArcSetin Tournaments( k -FAST) problem.Inthispaperweobtainalinearvertexkernel fork -FAST .Thatis,wegiveapolynomialtimealgorithmwhichgivenaninput instanceTtok -FAST obtainsanequivalentinstanceT 0 onO(k)vertices.Infact, givenanyfixed>0,thekernelizedinstancehasatmost(2+)kvertices.Our resultimprovesthepreviousknownboundofO(k 2 )onthekernelsizefork -FAST . Ourkernelizationalgorithmsolvestheproblemonasubclassoftournamentsin polynomialtimeandusesaknownpolynomialtimeapproximationschemefork - . TSAF 1Introduction GivenadirectedgraphG=(V,A)onnverticesandanintegerparameterk,the FeedbackArcSet problemaskswhetherthegivendigraphhasasetofkarcswhose removalresultsinanacyclicdirectedgraph.Inthispaper,weconsiderthisproblemin aspecialclassofdirectedgraphs,tournaments.AtournamentT=(V,A)isadirected graphinwhichthereisexactlyonedirectedarcbetweeneverypairofvertices.More formallytheproblemweconsiderisdefinedasfollows. k -FeedbackArcSetinTournaments (k -FAST ):Givenatournament T=(V,A)andapositiveintegerk,doesthereexistasubsetF ⊆ Aofat mostkarcswhoseremovalmakesTacyclic. Intheweightedversionofk -FAST ,wearealsogivenintegerweights(eachweight isatleastone)onthearcsandtheobjectiveistofindafeedbackarcsetofweightat mostk.Thisproblemiscalledk -WeightedFeedbackArcSetinTournaments ( k -WFAST) . ∗ LIRMM–Universite´deMontpellier2,CNRS,161rueAda,34392Montpellier,France. { bessy|paul|perez|thomasse } @lirmm.fr † DepartmentofInformatics,UniversityofBergen,N-5020Bergen,Norway. { fedor.fomin|saket.saurabh } @ii.uib.no ‡ CentrodeModelamientoMatema´tico,UniversidaddeChile,8370459SantiagodeChile. [e-mail] 1