Subexponential parameterized algorithm for computing the cutwidth of a semi-complete digraph

TL;DR

This paper introduces a subexponential parameterized algorithm for computing the cutwidth of semi-complete digraphs, significantly improving efficiency over previous algorithms and enabling faster solutions for related layout problems.

Contribution

It presents a novel subexponential algorithm for cutwidth in semi-complete digraphs and applies techniques to improve algorithms for Feedback Arc Set and Linear Arrangement problems.

Findings

Algorithm for cutwidth runs in 2^{O(√(k log k))} * n^{O(1)} time.

New simpler algorithm for Feedback Arc Set in tournaments with 2^{c√k} * n^{O(1)} time.

Linear Arrangement problem solved in 2^{O(k^{1/3} √log k)} * n^{O(1)} time.

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})} * 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}} * n^{O(1)}, where c = 2\pi / \sqrt(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(\sqrt{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} \sqrt{\log k})} * n^{O(1)} time, where k is the target cost of the ordering.