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

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!