Improved Coresets for Clustering with Capacity and Fairness Constraints

TL;DR

This paper presents an improved algorithm for constructing small coresets for capacitated and fair clustering problems, reducing computational complexity and coreset size compared to previous methods.

Contribution

It introduces a near-linear time algorithm with smaller coreset sizes for capacitated and fair clustering, improving hierarchical sampling by adaptive selection based on clustering cost.

Findings

Reduces coreset size for capacitated clustering by a factor of $k \\varepsilon^{-z}$.

Achieves near-linear time construction of $ ilde{O}(k^2\\\varepsilon^{-2z-2})$-sized coresets.

Simplifies analysis by decreasing the effective centers from $ ilde{O}(k^2\\\varepsilon^{-z})$ to $O(k \\\log \\\varepsilon^{-1})$.

Abstract

We study coresets for clustering with capacity and fairness constraints. Our main result is a near-linear time algorithm to construct $\tilde{O}(k^2\varepsilon^{-2z-2})$-sized $\varepsilon$-coresets for capacitated $(k,z)$-clustering which improves a recent $\tilde{O}(k^3\varepsilon^{-3z-2})$ bound by [BCAJ+22, HJLW23]. As a corollary, we also save a factor of $k \varepsilon^{-z}$ on the coreset size for fair $(k,z)$-clustering compared to them. We fundamentally improve the hierarchical uniform sampling framework of [BCAJ+22] by adaptively selecting sample size on each ring instance, proportional to its clustering cost to an optimal solution. Our analysis relies on a key geometric observation that reduces the number of total ``effective centers" from [BCAJ+22]'s $\tilde{O}(k^2\varepsilon^{-z})$ to merely $O(k\log \varepsilon^{-1})$ by being able to ``ignore'' all center points that are too far or too close to the ring center.