Kernels for feedback arc set in tournaments

  • Authors:

  • Stéphane Bessy;Fedor V. Fomin;Serge Gaspers;Christophe Paul;Anthony Perez;Saket Saurabh;Stéphan Thomassé

  • Affiliations:

  • LIRMM - Université Montpellier 2, CNRS, 161 rue Ada, 34932 Montpellier, France;Department of Informatics, University of Bergen, N-5020 Bergen, Norway;CMM, Universidad de Chile, Av. Blanco Encalada 2120, 8370459 Santiago de Chile, Chile;LIRMM - Université Montpellier 2, CNRS, 161 rue Ada, 34932 Montpellier, France;LIRMM - Université Montpellier 2, CNRS, 161 rue Ada, 34932 Montpellier, France;The Institute of Mathematical Sciences, Chennai 600 113, India;LIRMM - Université Montpellier 2, CNRS, 161 rue Ada, 34932 Montpellier, France

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2011

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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 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 @e0, the kernelized instance has at most (2+@e)k vertices. Our result improves the previous known bound of O(k^2) 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.