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

TL;DR

This paper introduces significantly faster fixed parameter algorithms for three ranking and tournament problems, achieving improvements over previous methods and enabling polynomial-time solutions for certain large instances.

Contribution

The authors develop new fixed parameter algorithms with improved runtimes for Kemeny rank aggregation, feedback arc set tournament, and betweenness tournament problems.

Findings

Achieved runtime O*(2^O(sqrt{OPT})) for Kemeny rank aggregation.

Improved feedback arc set tournament algorithm to runtime O*(2^O(sqrt{OPT})).

Enabled polynomial-time solutions for instances with OPT as large as n(log n)^2.

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 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})) (Alon, Lokshtanov and Saurabh 2009). For betweenness tournament we give an algorithm with runtime O*(2^O(sqrt{OPT/n})), where n is the number of vertices and OPT is the optimal cost. This improves on the previously known O*(OPT^O(OPT^{1/3}))$ (Saurabh 2009), especially when OPT is small. Unusually we can solve instances with OPT as large as n (log n)^2 in polynomial time!