2-Approximating Feedback Vertex Set in Tournaments

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. We consider the Feedback Vertex Set problem in tournaments. Here the input is a tournament T and a weight function w: V(T) → ℕ and the task is to find a feedback vertex set S in T minimizing w(S) = ΣvϵSw(v). Rounding optimal solutions to the natural LP-relaxation of this problem yields a simple 3-approximation algorithm. This has been improved to 2.5 by Cai et al. [SICOMP 2000], and subsequently to 7/3 by Mnich et al. [ESA 2016]. In this paper we give the first polynomial time factor 2 approximation algorithm for this problem. Assuming the Unique Games conjecture, this is the best possible approximation ratio achievable in polynomial time.