Faster Algorithms for Feedback Arc Set Tournament, Kemeny Rank Aggregation and Betweenness Tournament

Abstract

We study fixed parameter algorithms for three problems: Kemeny rank aggregation, feedback arc set tournament, and betweenness tournament. For Kemeny rank aggregation we give an algorithm with runtime $O^*(2^{O(\sqrt{OPT})})$, where n is the number of candidates, $OPT \le \binom{n}{2}$ is the cost of the optimal ranking, and O ^*(·) hides polynomial factors. This is a dramatic improvement on the previously best known runtime of O ^*(2^O(OPT)). For feedback arc set tournament we give an algorithm with runtime $O^*(2^{O(\sqrt{OPT})})$, an improvement on the previously best known $O^*(OPT^{O(\sqrt{OPT})})$ [4]. For betweenness tournament we give an algorithm with runtime $O^*(2^{O(\sqrt{OPT/n})})$, where n is the number of vertices and $OPT \le \binom{n}{3}$ is the optimal cost. This improves on the previously known $O^*(OPT^{O(OPT^{1/3})})$ [28], especially when OPT is small. Unusually we can solve instances with OPT as large as n (logn)² in polynomial time!

A preliminary version of this work appeared in version 1 of the arXiv preprint Karpinski and Schudy [23].