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

2026-02-24

AI Summary

This study investigates the computational complexity of two extensions of the Spearman footrule voting rule that incorporate task lengths or weights, focusing on scenarios with a small number of voters. Employing techniques from computational complexity theory, reduction methods, and combinatorial optimization, the work establishes for the first time that the first extension becomes NP-hard already with three voters, while the second extension is NP-hard with four voters; both problems, however, remain solvable in polynomial time when restricted to two voters. These results precisely delineate the complexity boundaries of these collective ranking problems in the regime of minimal electorate size, thereby filling a critical gap in the complexity landscape of related scheduling and ranking problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Game Theory and Economic Paradigms: Social Choice / Voting
• Machine Learning: Learning Preferences or Rankings

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

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.

Problem

• Spearman footrule
• NP-hard
• collective scheduling
• weighted candidates
• computational complexity

Innovation

• Spearman footrule
• NP-hardness
• collective scheduling
• weighted voting
• computational complexity