Approximating Fair $k$-Min-Sum-Radii in $\mathbb{R}^d$

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.

Coresets for constrained k-median and k-means clustering in low dimensional Euclidean space

We study (Euclidean) $k$-median and $k$-means with constraints in the streaming model. There have been recent efforts to design unified algorithms to solve constrained $k$-means problems without using knowledge of the specific constraint at hand aside from mild assumptions like the polynomial computability of feasibility under the constraint (compute if a clustering satisfies the constraint) or the presence of an efficient assignment oracle (given a set of centers, produce an optimal assignment of points to the centers which satisfies the constraint). These algorithms have a running time exponential in $k$, but can be applied to a wide range of constraints. We demonstrate that a technique proposed in 2019 for solving a specific constrained streaming $k$-means problem, namely fair $k$-means clustering, actually implies streaming algorithms for all these constraints. These work for low dimensional Euclidean space. [Note that there are more algorithms for streaming fair $k$-means today, in particular they exist for high dimensional spaces now as well.]