Subexponential Parameterized Algorithm for Computing the Cutwidth of a Semi-complete Digraph

Cutwidth of a digraph is a width measure introduced by Chudnovsky, Fradkin, and Seymour [4] in connection with development of a structural theory for tournaments, or more generally, for semi-complete digraphs. In this paper we provide an algorithm with running time 2 √ k log k) · n that tests whether the cutwidth of a given n-vertex semi-complete digraph is at most k, improving upon the currently fastest algorithm of the second author [18] that works in 2 · n time. As a byproduct, we obtain a new algorithm for Feedback Arc Set in tournaments (FAST) with running time 2 √ k · n, where c = 2π √ 3·ln 2 ≤ 5.24, that is simpler than the algorithms of Feige [9] and of Karpinski and Schudy[16], both also working in 2 √ k) ·nO(1) time. Our techniques can be applied also to other layout problems on semi-complete digraphs. We show that the Optimal Linear Arrangement problem, a close relative of Feedback Arc Set, can be solved in 2 1/3 · √ log k) · n time, where k is the target cost of the ordering.