{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8373","name":"Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Tournaments","abstract":"A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T\\S is acyclic. In this article we consider the FEEDBACK VERTEX SET problem in tournaments. Here the input is a tournament T and an integer k, and the task is to determine whether T has a feedback vertex set of size at most k. We give a new algorithm for FEEDBACK VERTEX SET IN TOURNAMENTS. The running time of our algorithm is upper-bounded by O(1.6181^k + n^{O(1)}) and by O(1.466^n). Thus our algorithm simultaneously improves over the fastest known parameterized algorithm for the problem by Dom et al. running in time O(2^kk^{O(1)} + n^{O(1)}), and the fastest known exact exponential-time algorithm by Gaspers and Mnich with running time O(1.674^n). On the way to proving our main result we prove a strengthening of a special case of a graph partitioning theorem due to Bollobas and Scott. In particular we show that the vertices of any undirected m-edge graph of maximum degree d can be colored white or black in such a way that for each of the two colors, the number of edges with both endpoints of that color is between m\/4-d\/2 and m\/4+d\/2.","keywords":["Parameterized algorithms","Exact algorithms","Feedback vertex set","Tour- naments","Graph partitions"],"author":[{"@type":"Person","name":"Kumar, Mithilesh","givenName":"Mithilesh","familyName":"Kumar"},{"@type":"Person","name":"Lokshtanov, Daniel","givenName":"Daniel","familyName":"Lokshtanov"}],"position":49,"pageStart":"49:1","pageEnd":"49:13","dateCreated":"2016-02-16","datePublished":"2016-02-16","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kumar, Mithilesh","givenName":"Mithilesh","familyName":"Kumar"},{"@type":"Person","name":"Lokshtanov, Daniel","givenName":"Daniel","familyName":"Lokshtanov"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2016.49","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/dx.doi.org\/10.1002\/jgt.v46:2","isPartOf":{"@type":"PublicationVolume","@id":"#volume6250","volumeNumber":47,"name":"33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)","dateCreated":"2016-02-16","datePublished":"2016-02-16","editor":[{"@type":"Person","name":"Ollinger, Nicolas","givenName":"Nicolas","familyName":"Ollinger"},{"@type":"Person","name":"Vollmer, Heribert","givenName":"Heribert","familyName":"Vollmer"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8373","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6250"}}}