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Kernels for Feedback Arc Set In Tournaments

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 ϵ > 0, the kernelized instance has at most (2 + ϵ)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.
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https://hal-lirmm.ccsd.cnrs.fr/lirmm-00432668
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Stéphane Bessy, Fedor V. Fomin, Serge Gaspers, Christophe Paul, Anthony Perez, et al.. Kernels for Feedback Arc Set In Tournaments. IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, Dec 2009, IIT Kanpur, India. pp.37-47, ⟨10.4230/LIPIcs.FSTTCS.2009.2305⟩. ⟨lirmm-00432668⟩

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