Two NP-hard Extensions of the Spearman Footrule even for a Small Constant Number of Voters

TL;DR

This paper proves that two natural extensions of the Spearman footrule voting rule are NP-hard even with a small constant number of voters, highlighting computational challenges in these voting models.

Contribution

The paper establishes NP-hardness results for two extensions of the Spearman footrule with a small number of voters, which were previously known to be polynomial-time solvable.

Findings

NP-hard with 3 voters for the first extension

NP-hard with 4 voters for the second extension

Polynomial-time solvable for 2 voters in both extensions

Abstract

The Spearman footrule is a voting rule that takes as input voter preferences expressed as rankings. It outputs a ranking that minimizes the sum of the absolute differences between the position of each candidate in the ranking and in the voters' preferences. In this paper, we study the computational complexity of two extensions of the Spearman footrule when the number of voters is a small constant. The first extension, introduced by Pascual et al. (2018), arises from the collective scheduling problem and treats candidates, referred to as tasks in their model, as having associated lengths. The second extension, proposed by Kumar and Vassilvitskii (2010), assigns weights to candidates; these weights serve both as lengths, as in the collective scheduling model, and as coefficients in the objective function to be minimized. Although computing a ranking under the standard Spearman footrule is polynomial-time solvable, we demonstrate that the first extension is NP-hard with as few as 3 voters, and the second extension is NP-hard with as few as 4 voters. Both extensions are polynomial-time solvable for 2 voters.