On Optimal Coreset Construction for Euclidean $(k,z)$-Clustering

Abstract

Constructing small-sized coresets for various clustering problems in different metric spaces has attracted significant attention for the past decade. A central problem in the coreset literature is to understand what is the best possible coreset size for $(k,z)$-clustering in Euclidean space. While there has been significant progress in the problem, there is still a gap between the state-of-the-art upper and lower bounds. For instance, the best known upper bound for $k$-means ($z=2$) is $\min \{O(k^{3/2}{\epsilon }^{-2}),O(k{\epsilon }^{-4})\}$ [1,2], while the best known lower bound is $\Omega (k{\epsilon }^{-2})$ [1]. In this paper, we make significant progress on both upper and lower bounds. For a large range of parameters (i.e., $\epsilon ,k$), we have a complete understanding of the optimal coreset size. In particular, we obtain the following results: (1) We present a new coreset lower bound $\Omega (k{\epsilon }^{-z-2})$ for Euclidean $(k,z)$-clustering when $\epsilon \ge \Omega (k^{-1/(z+2)})$. In view of the prior upper bound ${\tilde{O}}_z(k{\epsilon }^{-z-2})$ [1], the bound is optimal. The new lower bound also implies improved lower bounds for $(k,z)$-clustering in doubling metrics. (2) For the upper bound, we provide efficient coreset construction algorithms for $(k,z)$-clustering with improved or optimal coreset sizes in several metric spaces. In particular, we provide an ${\tilde{O}}_z(k^{\frac{2z+2}{z+2}}{\epsilon }^{-2})$-sized coreset, with a unfied analysis, for $(k,z)$-clustering for all $z\ge 1$ in Euclidean space. [1] Cohen-Addad, Larsen, Saulpic, Schwiegelshohn. STOC'22. [2] Cohen-Addad, Larsen, Saulpic, Schwiegelshohn, Sheikh-Omar, NeurIPS'22.

Accepted by STOC '24

Cited by in corpus (1)

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