 License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2016.49
URN: urn:nbn:de:0030-drops-57501
URL: https://drops.dagstuhl.de/opus/volltexte/2016/5750/
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### Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Tournaments

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### 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.

### BibTeX - Entry

```@InProceedings{kumar_et_al:LIPIcs:2016:5750,
author =	{Mithilesh Kumar and Daniel Lokshtanov},
title =	{{Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Tournaments}},
booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
pages =	{49:1--49:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-001-9},
ISSN =	{1868-8969},
year =	{2016},
volume =	{47},
editor =	{Nicolas Ollinger and Heribert Vollmer},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
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