On Tight Robust Coresets for $k$-Medians Clustering

TL;DR

This paper develops tight coreset constructions for robust $k$-medians clustering with outliers across various metric spaces, achieving near-optimal sizes and extending to $(k,z)$-clustering with improved bounds.

Contribution

It introduces new coreset constructions with optimal or near-optimal sizes for robust $k$-medians and $(k,z)$-clustering in multiple metric spaces, using novel dataset decomposition techniques.

Findings

Coreset size $O(m) + ilde{O}(kd ext{...})$ for bounded VC/doubling spaces

Improved coreset size $O(m ext{...}) + ilde{O}( ext{min}\{k^{4/3} ext{...},k ext{...} ight)$ for Euclidean spaces

Extensions to robust $(k,z)$-clustering with improved bounds

Abstract

This paper considers coresets for the robust $k$-medians problem with $m$ outliers, and new constructions in various metric spaces are obtained. Specifically, for metric spaces with a bounded VC or doubling dimension $d$, the coreset size is $O(m) + \tilde{O}(kd\varepsilon^{-2})$, which is optimal up to logarithmic factors. For Euclidean spaces, the coreset size is $O(m\varepsilon^{-1}) + \tilde{O}(\min\{k^{4/3}\varepsilon^{-2},k\varepsilon^{-3}\})$, improving upon a recent result by Jiang and Lou (ICALP 2025). These results also extend to robust $(k,z)$-clustering, yielding, for VC and doubling dimension, a coreset size of $O(m) + \tilde{O}(kd\varepsilon^{-2z})$ with the optimal linear dependence on $m$. This extended result improves upon the earlier work of Huang et al. (SODA 2025). The techniques introduce novel dataset decompositions, enabling chaining arguments to be applied jointly across multiple components.