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

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 Chebyschev inequality applied to the parity matrix of subset-intersection. Lemma 1 Let z be an integer. We denote by A the 2 × 2 matrix whose rows and columns are indexed by the subsets Fi of {1, . . . , z} (in any order) and whose entries are aij = (−1)|(Fi∩Fj)|. For any subset J of r columns, we have: