Optimally Selecting Representative Agents from a Metric Space

2026-08-29

AI Summary

本文解决了公平聚类问题，通过选择k个中心点来公平代表位于度量空间中的代理。使用Scarf定理证明了2-Droop核心的存在性，并解决了β-多数问题。

Abstract

This paper studies the problem of proportionally fair clustering, where the goal is to select $k$ ``centers'' from a metric space that fairly represent a set of agents who also lie in the metric space. Specifically, we focus on finding a clustering satisfying a fairness property known as the Droop core. In the practical special case in which the set of feasible center locations contains every agent location, the previous best-known result guaranteed a $(1 + \sqrt{2})$-approximation of the Droop core, while the best-known lower bound was $2$. In this paper, we show that this lower bound is tight and that a clustering in the $2$-Droop core always exists. Further, we show that such a clustering can be achieved by only selecting centers from locations in the metric space where an agent resides. We establish this using Scarf's theorem guaranteeing a nonempty core for balanced non-transferable utility games. This result has several interesting corollaries. Most notably, it resolves the $β$-plurality problem of Aronov et al. [2021] for general metric spaces. The main result of this paper was generated by $\mathtt{ChatGPT}$-$\mathtt{5.6}$-$\mathtt{Sol}$ through a series of interactions with the authors. The authors of this paper verified the generated proof and rewrote it for clarity.

Problem

• proportionally fair clustering
• Droop core
• metric space

Innovation

• proportionally fair clustering
• Droop core
• metric space
• Scarf's theorem
• β-plurality problem