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 semicomplete digraphs. In this paper we provide an algorithm with running time 2~(O(~(1/2)k log k)) _n~(O(1)) 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~(O(k)) · n~2 time. As a byproduct, we obtain a new algorithm for Feedback Arc Set in tournaments (FAST) with running time 2~(c~(1/2)k) · n~(O(1)), where c = 2π/~(1/2)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~(O(~(1/2)k)) ·n~(O(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~(O(k~(1/3)·~(1/2)log k) · n~(O(1)) time, where k is the target cost of the ordering.
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