Approximating Fair $k$-Min-Sum-Radii in Euclidean Space

TL;DR

The paper addresses the problem of minimizing the sum of cluster radii in a $k$-clustering (the $k$-min-sum-radii problem) under mergeable fairness constraints in Euclidean space of arbitrary dimension with constant $k$. It develops a PTAS by extending the idea behind the $k$-center PTAS of Bădoiu et al. to the MSR objective, introducing the concepts of $oldsymbol{ extgamma}$-separated and $oldsymbol{ extepsilon}$-balanced coverings to structure the solution. The approach relies on radii guessing linked to a constrained $k$-center approximation, plus constructing approximate MEBs and using a Selection procedure to iteratively cover the point set while preserving the mergeable constraints. The main contributions include the first PTAS for fair $k$-MSR with mergeable constraints, a detailed framework for handling various fairness notions, and a runtime that scales favorably when a constant-factor constrained $k$-center approximation exists. This work advances fair clustering by providing provable approximation guarantees for a nuanced objective that balances per-cluster radii while respecting group fairness constraints, with potential practical impact in resource placement and network design.

Abstract

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.

Figures (2)

• Figure 1: Left side: An example where $k$-min-sum-radii rather opens one cluster than eleven. Right side: An example where the cheapest $k=2$-clustering keeps $c_1$ as a singleton rather than combining it with $x$, despite the fact that $c_1$ is closer to $x$ than the center $c_2$ of the big cluster. It is cheaper to not assign the blue point to the orange point even though that would be a closer center.
• Figure 2: Example run of the algorithm for $\varepsilon=0.2$. In every iteration, the purple points depict the points that were already chosen by the algorithm. For reference, the black circles represent the true minimum enclosing balls of the subsets $S_i$. These are not computed in the algorithm, but only the $(1+\varepsilon)$-approximations of these, depicted in orange. The blue circles enclose the areas which we ignore when sampling new points. That is, for singleton clusters, it is $B(s_j, \frac{\varepsilon}{1+\varepsilon}\widetilde{r_j})$, and once the algorithm found two points from a cluster, it computes an approximate MEB and enlarges it by a factor of $\gamma = 1+\varepsilon + 2\sqrt{\varepsilon}$ to obtain the new blue ball.

Theorems & Definitions (34)

• Theorem 1
• Definition 2
• Definition 3
• Lemma 4
• proof
• Lemma 4
• Definition 5
• Lemma 5
• Lemma 6
• proof