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

Martin Durand

TL;DR

Although computing a ranking under the standard Spearman footrule is polynomial-time solvable, it is demonstrated 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.

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.

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

TL;DR

Although computing a ranking under the standard Spearman footrule is polynomial-time solvable, it is demonstrated 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.

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.
Paper Structure (8 sections, 5 theorems, 8 equations, 6 figures)

This paper contains 8 sections, 5 theorems, 8 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations.
  4. Computational Complexity.
  5. Case (1)
  6. Case (2)
  7. Case (3)
  8. Conclusion

Key Result

Theorem 1

The $\Sigma$D-Dec problem is NP-hard, even when there are only 4 voters.

Figures (6)

  • Figure 1: Preferences of the voters in the four voters reduction. Dashed tasks means that there are as many unit tasks as the size of a target triplet in the 3-Partition instance. Tasks from all sets are always scheduled in the same order by the voters.
  • Figure 2: High-level view of the preferred schedules of the three voters.
  • Figure 3: Illustration of Case (3). $C_j$ (resp. $C_{j+1}$) denotes the completion time of $a\xspace_j^i$ (resp. $a\xspace_{j+1}^i$) in the voters' preferences. A light blue arrow indicates a reduction in deviation when the corresponding task is moved in the direction of the arrow, while a dashed dark red arrow indicates an increase in deviation.
  • Figure 5: Structure of an existing optimal solution. Arrows indicate that the separator sets, from $A_{1}$ to $A_{q\xspace+1}$, complete close to their completion times in the voters' preferences.
  • Figure 6: Infeasible ideal schedule.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof