The Minimum Feedback Arc Set Problem is NP-Hard for Tournaments

Abstract

Answering a question of Bang-Jensen and Thomassen [4], we prove that the minimum feedback arc set problem is NP-hard for tournaments. A feedback arc set (fas) in a digraph D = (V , A) is a set F of arcs such that D \ F is acyclic. The size of a minimum feedback arc set of D is denoted by mfas(D). A classical result of Lawler and Karp [5] asserts that finding a minimum feedback arc set in a digraph is NP-hard. Bang-Jensen and Thomassen [4] conjectured that finding a minimum fas in a tournament is also NP-hard. A very close answer was given by Ailon, Charikar and Newman in [1], where they prove that the problem is NP-hard under randomized reductions. Our approach is similar, but the reduction we use is simpler and therefore easily derandomized via parity-check matrices (see Alon and Spencer [3, p. 255]). Finally we prove that the minimum fas for tournaments is polynomially equivalent to the minimum fas for digraphs, and thus NP-hard. The following lemma is just Chebyshev inequality applied to the parity matrix of subset-intersection. Lemma 1. Let z be an integer. We denote by A the 2 z × 2 z matrix whose rows and columns are indexed by the subsets F i of {1,. .. , z} (in any order) and whose entries are a ij = (−1) |(F i ∩F j)|. For any subset J of r columns, we have 2 z i=1 j∈J a ij 2 z √ r.

Some Parameterized Problems On Digraphs

“…Bang-Jensen and Thomassen [5] conjectured that FAS is N P-complete even for tournaments. This conjecture was proved independently by at least four groups of researchers [1,2,8,9]. Interestingly, FAS is polynomial time solvable for planar digraphs [4,35] and trivially polynomial time solvable for undirected graphs.…”

Ranking Tournaments: Local Search and a New Algorithm

“…Dwork et al [10] prove the Rank Aggregation problem is NP-hard, even with as few as 4 input rankings. Recently, the FAS problem has been shown to be NP-hard even on unweighted tournaments [7]. Given this, it is natural to ask for an approximation algorithm which runs in polynomial time, yet is guaranteed to differ in cost from the optimal solution by a small factor.…”

A Polynomial Time Approximation Scheme for k-Consensus Clustering

“…The latter are frequently easier to approximate (see below), and we show that this dichotomy extends to metric variants. There has been considerable recent interest in the approximability of the Consensus Clustering [1] and Rank Aggregation problems [6,20]. In particular, Bonizzoni et al showed that Consensus Clustering is APX-hard [5]; our paper is a response to this result.…”