The $k$-center problem is a classical clustering problem in which one is asked to find a partitioning of a point set $P$ into $k$ clusters such that the maximum radius of any cluster is minimized. It is well-studied. But what if we add up the radii of the clusters instead of only considering the cluster with maximum radius? This natural variant is called the $k$-min-sum-radii problem. It has become the subject of more and more interest in recent years, inspiring the development of approximation algorithms for the $k$-min-sum-radii problem in its plain version as well as in constrained settings. We study the problem for Euclidean spaces $\mathbb{R}^d$ of arbitrary dimension but assume the number $k$ of clusters to be constant. In this case, a PTAS for the problem is known (see Bandyapadhyay, Lochet and Saurabh, SoCG, 2023). Our aim is to extend the knowledge base for $k$-min-sum-radii to the domain of fair clustering. We study several group fairness constraints, such as the one introduced by Chierichetti et al. (NeurIPS, 2017). In this model, input points have an additional attribute (e.g., colors such as red and blue), and clusters have to preserve the ratio between different attribute values (e.g., have the same fraction of red and blue points as the ground set). Different variants of this general idea have been studied in the literature. To the best of our knowledge, no approximative results for the fair $k$-min-sum-radii problem are known, despite the immense amount of work on the related fair $k$-center problem. We propose a PTAS for the fair $k$-min-sum-radii problem in Euclidean spaces of arbitrary dimension for the case of constant $k$. To the best of our knowledge, this is the first PTAS for the problem. It works for different notions of group fairness.

翻译：$k$-中心问题是一个经典的聚类问题，其目标是将点集$P$划分为$k$个簇，使得任意簇的最大半径最小化。该问题已被广泛研究。但如果我们考虑所有簇的半径之和，而不仅仅是最大半径的簇，会如何呢？这一自然变体被称为$k$-最小和半径问题。近年来该问题受到越来越多的关注，推动了针对其基本形式及受限场景的近似算法的发展。我们研究任意维度欧几里得空间$\mathbb{R}^d$中的该问题，并假设簇的数量$k$为常数。在此条件下，已知该问题存在多项式时间近似方案（PTAS）（参见 Bandyapadhyay、Lochet 和 Saurabh，SoCG，2023）。我们的目标是将$k$-最小和半径问题的知识体系扩展到公平聚类领域。我们研究了多种群体公平约束，例如由 Chierichetti 等人（NeurIPS，2017）提出的模型。在此模型中，输入点具有额外属性（例如颜色属性如红与蓝），且簇必须保持不同属性值之间的比例（例如，与整体数据集具有相同的红蓝点比例）。文献中已对此基本思想的不同变体进行了研究。据我们所知，尽管对相关的公平$k$-中心问题已有大量研究，但公平$k$-最小和半径问题尚未有近似结果发表。针对常数$k$的情形，我们为任意维度欧几里得空间中的公平$k$-最小和半径问题提出了一个PTAS。据我们所知，这是该问题的首个PTAS，适用于多种群体公平性定义。