Fixed-Parameter Algorithms for Computing Kemeny Scores - Theory and Practice

TL;DR

This paper explores fixed-parameter algorithms for the NP-hard Kemeny consensus problem, demonstrating their practical effectiveness on real data and improving theoretical bounds for certain parameters.

Contribution

It introduces an improved fixed-parameter algorithm for the Kemeny consensus problem with better theoretical running time bounds.

Findings

Algorithms are practically useful on real-world data.

Fixed-parameter algorithms outperform general approaches.

Improved theoretical bounds for specific parameters.

Abstract

The central problem in this work is to compute a ranking of a set of elements which is "closest to" a given set of input rankings of the elements. We define "closest to" in an established way as having the minimum sum of Kendall-Tau distances to each input ranking. Unfortunately, the resulting problem Kemeny consensus is NP-hard for instances with n input rankings, n being an even integer greater than three. Nevertheless this problem plays a central role in many rank aggregation problems. It was shown that one can compute the corresponding Kemeny consensus list in f(k) + poly(n) time, being f(k) a computable function in one of the parameters "score of the consensus", "maximum distance between two input rankings", "number of candidates" and "average pairwise Kendall-Tau distance" and poly(n) a polynomial in the input size. This work will demonstrate the practical usefulness of the corresponding algorithms by applying them to randomly generated and several real-world data. Thus, we show that these fixed-parameter algorithms are not only of theoretical interest. In a more theoretical part of this work we will develop an improved fixed-parameter algorithm for the parameter "score of the consensus" having a better upper bound for the running time than previous algorithms.