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

Abstract

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 \(2^{O(\sqrt{k\log k})}\cdot n^{O(1)}\) 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 \(2^{c\sqrt{k}}\cdot n^{O(1)}\), where \(c=\frac{2\pi}{\sqrt{3}\cdot \ln 2}\leq 5.24\), that is simpler than the algorithms of Feige [9] and of Karpinski and Schudy [16], both also working in \(2^{O(\sqrt{k})}\cdot n^{O(1)}\) 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 \(2^{O(k^{1/3}\cdot\sqrt{\log k})}\cdot n^{O(1)}\) time, where k is the target cost of the ordering.