When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2009.2305
URN: urn:nbn:de:0030-drops-23055
URL: http://drops.dagstuhl.de/opus/volltexte/2009/2305/
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### Kernels for Feedback Arc Set In Tournaments

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### 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 {\sc Feedback Arc Set} problem asks whether thegiven digraph has a set of $k$ arcs whose removal results in an acyclicdigraph. The {\sc Feedback Arc Set} problem restricted to tournaments is knownas the {\sc $k$-Feedback Arc Set in Tournaments ($k$-FAST)} problem. In thispaper we obtain a linear vertex kernel for \FAST{}. That is, we give apolynomial time algorithm which given an input instance $T$ to \FAST{} obtains an equivalent instance $T'$ on $O(k)$ vertices. In fact, given any fixed $\epsilon > 0$, the kernelized instance has at most $(2 + \epsilon)k$ vertices.Our result improves the previous known bound of $O(k^2)$ on the kernel size for\FAST{}. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for \FAST.

### BibTeX - Entry

@InProceedings{bessy_et_al:LIPIcs:2009:2305,
author =	{St{\'e}phane Bessy and Fedor V. Fomin and Serge Gaspers and Christophe Paul and Anthony Perez and Saket Saurabh and St{\'e}phan Thomass{\'e}},
title =	{{Kernels for Feedback Arc Set In Tournaments}},
booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
pages =	{37--47},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-13-2},
ISSN =	{1868-8969},
year =	{2009},
volume =	{4},
editor =	{Ravi Kannan and K. Narayan Kumar},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},