从度量空间中最优选择代表性智能体

中文摘要 AI 辅助

本文研究比例公平聚类问题，目标是从度量空间中选取k个“中心”，公平代表同样位于该空间中的一组智能体。具体而言，我们聚焦于满足名为Droop核心的公平属性的聚类。在可行中心位置包含所有智能体位置的实际特殊情形中，此前已知的最佳结果保证了Droop核心的(1+√2)近似比，而最佳已知下界为2。本文中，我们证明该下界是紧的，且2-Droop核心中始终存在一个聚类；进一步表明，仅从智能体所在的度量空间位置选取中心即可实现此类聚类。我们利用Scarf定理（该定理保证平衡型非转移效用博弈的核心非空）建立这一结论。该结果有若干有趣推论，最值得注意的是，它解决了Aronov等人[2021]提出的针对一般度量空间的β-多数问题。本文的主要结果由ChatGPT-5.6-Sol通过与作者的一系列交互生成，作者验证了生成的证明并为清晰性进行了重写。

英文摘要

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.

发表机构

• University of Toronto（多伦多大学）
• TU Clausthal（克劳斯塔尔工业大学）
• The University of Tokyo（东京大学）

机构由 AI 辅助整理，请以论文原文为准。