arXiv:2008.08779v1 [math.CO] 20 Aug 2020 A SIMPLE 7/3-APPROXIMATION ALGORITHM FOR FEEDBACK VERTEX SET IN TOURNAMENTS MANUEL APRILE, MATTHEW DRESCHER, SAMUEL FIORINI, AND TONY HUYNH Abstract. We show that performing just one round of the Sherali-Adams hierarchy gives an easy 7/3-approximation algorithm for the Feedback Vertex Set (FVST) prob- lem in tournaments. This matches the best deterministic approximation algorithm for FVST due to Mnich, Williams, and Végh [9], and is a significant simplification and runtime improvement of their approach. 1. Introduction A feedback vertex set (FVS) of a tournament T is a set X of vertices such that T − X is acyclic. Given a tournament T and (vertex) weights w : V (T ) → Q ≥0 , the Feedback Vertex Set (FVST) problem asks to find a feedback vertex set X such that w(X) := ∑ x∈X w(x) is minimum. This problem has numerous applications, for example in determining election winners in social choice theory [2]. We let OPT(T,w) be the minimum weight of a feedback vertex set of the weighted tournament (T,w). An α-approximation algorithm for FVST is a polynomial-time algo- rithm computing a feedback vertex set X with w(X) ≤ α · OPT(T,w). Note that a tournament is acyclic if and only if it does not contain a directed tri- angle. Therefore, the following is an easy 3-approximation algorithm for FVST in the unweighted case (the general case follows for instance from the local ratio technique [4]). If T is acyclic, then ∅ is an FVS, and we are done. Otherwise, we find a directed triangle abc in T and put all its vertices into the FVS. We then replace T by T −{a, b, c} and recurse. State of the Art. The first non-trivial approximation algorithm for FVST was a 5/2- approximation algorithm by Cai, Deng, and Zang [3]. Cai et al.’s approach is polyhedral. It is based on the fact that the basic LP relaxation of FVST is integral whenever the input tournament avoids certain subtournaments, see the next paragraphs for details. Let T be a tournament and △(T ) denote the collection of all {a, b, c}⊆ V (T ) that induce a directed triangle in T . The basic relaxation for T is the polytope P (T ) := {x ∈ [0, 1] V (T ) |∀{a, b, c}∈△(T ): x a + x b + x c ≥ 1}. Let T 5 be the set of tournaments on 5 vertices where the minimum FVS has size 2. Up to isomorphism, |T 5 | =3 (see [3]). We say that T is T 5 -free if no subtournament of T is isomorphic to a member of T 5 . More generally, let T be a collection of tournaments. A Date : August 21, 2020. This project was supported by ERC Consolidator Grant 615640-ForEFront. Samuel Fiorini and Manuel Aprile are also supported by FNRS grant T008720F-35293308-BD-OCP. Tony Huynh is also supported by the Australian Research Council. 1