Parameterized Aspects of Distinct Kemeny Rank Aggregation

TL;DR

This paper explores the parameterized complexity of computing Kemeny rankings, analyzing relationships among parameters, and developing fixed-parameter tractable algorithms, including approximation methods, for efficient rank aggregation.

Contribution

It provides a comprehensive analysis of parameter relationships and introduces FPT algorithms for computing and approximating all distinct Kemeny rankings.

Findings

Relationships among parameters are both theoretical and empirical.

FPT algorithms can compute all distinct Kemeny rankings efficiently.

FPT approximation algorithms are developed for various parameters.

Abstract

The Kemeny method is one of the popular tools for rank aggregation. However, computing an optimal Kemeny ranking is NP-hard. Consequently, the computational task of finding a Kemeny ranking has been studied under the lens of parameterized complexity with respect to many parameters. We first present a comprehensive relationship, both theoretical and empirical, among these parameters. Further, we study the problem of computing all distinct Kemeny rankings under the lens of parameterized complexity. We consider the target Kemeny score, number of candidates, average distance of input rankings, maximum range of any candidate, and unanimity width as our parameters. For all these parameters, we already have FPT algorithms. We find that any desirable number of Kemeny rankings can also be found without substantial increase in running time. We also present FPT approximation algorithms for Kemeny rank aggregation with respect to these parameters.