Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Proportional Clustering, the $β$-Plurality Problem, and Metric Distortion

Leon Kellerhals, Jannik Peters

TL;DR

This paper establishes a fundamental link between proportional clustering under the Droop quota and the $\beta$-plurality problem, showing equivalence when the number of centers is fixed at $k=1$ and connecting both to metric distortion. It demonstrates that the plurality veto rule can obtain $$(\sqrt{5}-2)$$-plurality points using only ordinal information, and uses this to derive $$(2+\sqrt{5})$$-proportional clusterings from ordinal data, addressing an open question. The results suggest that methods developed for $\beta$-plurality translate to proportional clustering in the Droop setting, and highlight avenues for extending these techniques to larger $k$ and to different metric spaces. The work thus advances both fair clustering and ordinal approximation, with potential practical impact on elections, facility location, and multiwinner rules that rely on limited distance information.

Abstract

We show that the proportional clustering problem using the Droop quota for $k = 1$ is equivalent to the $β$-plurality problem. We also show that the Plurality Veto rule can be used to select ($\sqrt{5} - 2$)-plurality points using only ordinal information about the metric space and resolve an open question of Kalayci et al. (AAAI 2024) by proving that $(2+\sqrt{5})$-proportionally fair clusterings can be found using purely ordinal information.

Proportional Clustering, the $β$-Plurality Problem, and Metric Distortion

TL;DR

This paper establishes a fundamental link between proportional clustering under the Droop quota and the -plurality problem, showing equivalence when the number of centers is fixed at and connecting both to metric distortion. It demonstrates that the plurality veto rule can obtain -plurality points using only ordinal information, and uses this to derive -proportional clusterings from ordinal data, addressing an open question. The results suggest that methods developed for -plurality translate to proportional clustering in the Droop setting, and highlight avenues for extending these techniques to larger and to different metric spaces. The work thus advances both fair clustering and ordinal approximation, with potential practical impact on elections, facility location, and multiwinner rules that rely on limited distance information.

Abstract

We show that the proportional clustering problem using the Droop quota for is equivalent to the -plurality problem. We also show that the Plurality Veto rule can be used to select ()-plurality points using only ordinal information about the metric space and resolve an open question of Kalayci et al. (AAAI 2024) by proving that -proportionally fair clusterings can be found using purely ordinal information.

Paper Structure

This paper contains 11 sections, 7 theorems, 10 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proportional Clustering and Plurality Points
  3. Proportional clustering
  4. β-plurality points
  5. Connecting plurality points and proportionality
  6. Plurality Points and Distortion
  7. Plurality Points Using Only Ordinal Information
  8. Proportional Clusterings Using Only Ordinal Information
  9. Conclusion
  10. Acknowledgements
  11. A connection between rank-PJR and the core

Key Result

Theorem 1

Let $\ell \in \mathbb N$ with $\ell > \frac{n}{k+1}$. Then greedy capture returns an outcome satisfying $\alpha$-approximate $\ell$-proportionality, where $\alpha \le (1 + \sqrt{2})$ in general metric spaces and $\alpha \le 2$ in Euclidean spaces.

Theorems & Definitions (16)

  • Definition 1: $\alpha$-approximate $\ell$-proportionality
  • Theorem 1
  • Definition 2
  • Theorem 2
  • proof
  • Definition 3
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • ...and 6 more