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，要求将点集 $P$ 划分为 $k$ 个簇，使得所有簇的最大半径最小化。该问题已被广泛研究。然而，如果我们不是仅考虑最大半径的簇，而是将各簇的半径相加，结果会如何？这一自然变体被称为 $k$-最小和半径问题。近年来，该问题引起了越来越多的关注，推动了其在无约束版本及约束设置下的近似算法的发展。我们研究了任意维度欧几里得空间 $\mathbb{R}^d$ 中的该问题，但假设聚类数量 $k$ 为常数。在此情况下，已知该问题存在一个多项式时间近似方案（PTAS，见 Bandyapadhyay、Lochet 和 Saurabh，SoCG，2023）。我们的目标是拓展 $k$-最小和半径问题的知识基础，将其引入公平聚类领域。我们研究了多种群体公平约束，例如 Chierichetti 等人（NeurIPS，2017）提出的约束。在该模型中，输入点具有额外属性（例如红色和蓝色等颜色），且簇必须保持不同属性值之间的比例（例如，簇中红点和蓝点的比例与整体集合相同）。文献中已研究了这一总体思想的不同变体。据我们所知，尽管相关公平 $k$-中心问题已有大量研究，但关于公平 $k$-最小和半径问题尚无近似结果。我们针对任意维欧几里得空间中常数 $k$ 的情况，提出了公平 $k$-最小和半径问题的多项式时间近似方案。据我们所知，这是该问题的首个多项式时间近似方案，且适用于不同的群体公平概念。