Coresets for Constrained Clustering: General Assignment Constraints and Improved Size Bounds

TL;DR

This paper develops new coresets for constrained clustering problems, including capacity and fairness constraints, achieving smaller sizes and broader applicability than previous methods.

Contribution

It introduces a general framework for coresets under assignment constraints, improving size bounds and extending to fault-tolerant clustering across various metric spaces.

Findings

First $ ilde{O}(m + k^2 ext{poly}(rac{1}{ ext{epsilon}}))$ coreset for capacitated and fair $k$-Median with outliers.

Improved size bounds over prior work for constrained clustering problems.

Extended coresets to fault-tolerant clustering in diverse metric spaces.

Abstract

Designing small-sized \emph{coresets}, which approximately preserve the costs of the solutions for large datasets, has been an important research direction for the past decade. We consider coreset construction for a variety of general constrained clustering problems. We introduce a general class of assignment constraints, including capacity constraints on cluster centers, and assignment structure constraints for data points (modeled by a convex body $\mathcal{B}$). We give coresets for clustering problems with such general assignment constraints that significantly generalize and improve known results. Notable implications include the first $\varepsilon$-coreset for capacitated and fair $k$-Median with $m$ outliers in Euclidean spaces whose size is $\tilde{O}(m + k^2 \varepsilon^{-4})$, generalizing and improving upon the prior bounds in [Braverman et al., FOCS' 22; Huang et al., ICLR' 23] (for capacitated $k$-Median, the coreset size bound obtained in [Braverman et al., FOCS' 22] is $\tilde{O}(k^3 \varepsilon^{-6})$, and for $k$-Median with $m$ outliers, the coreset size bound obtained in [Huang et al., ICLR' 23]} is $\tilde{O}(m + k^3 \varepsilon^{-5})$), and the first $\epsilon$-coreset of size $\mathrm{poly}(k \varepsilon^{-1})$ for fault-tolerant clustering for various types of metric spaces.