Constructing small-sized coresets for various clustering problems in Euclidean 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 $\min \{O(k^{3/2} \varepsilon^{-2}),O(k \varepsilon^{-4})\}$ [1,2], while the best known lower bound is $\Omega(k\varepsilon^{-2})$ [1]. In this paper, we make significant progress on both upper and lower bounds. For a large range of parameters (i.e., $\varepsilon, k$), 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 $\Omega(k \varepsilon^{-z-2})$ for Euclidean $(k,z)$-clustering when $\varepsilon \geq \Omega(k^{-1/(z+2)})$. In view of the prior upper bound $\tilde{O}_z(k \varepsilon^{-z-2})$ [1], the bound is optimal. The new lower bound is surprising since $\Omega(k\varepsilon^{-2})$ [1] is ``conjectured" to be the correct bound in some recent works (see e.g., [1,2]]). (2) For the upper bound, we provide efficient coreset construction algorithms for Euclidean $(k,z)$-clustering with improved coreset sizes. In particular, we provide an $\tilde{O}_z(k^{\frac{2z+2}{z+2}} \varepsilon^{-2})$-sized coreset, with a unfied analysis, for $(k,z)$-clustering for all $z\geq 1$ in Euclidean space. [1] Cohen-Addad, Larsen, Saulpic, Schwiegelshohn. STOC'22. [2] Cohen-Addad, Larsen, Saulpic, Schwiegelshohn, Sheikh-Omar, NeurIPS'22.

翻译：在欧几里得空间中为各类聚类问题构造小规模核心集，在过去十年中引起了广泛关注。核心集研究中的一个核心问题是理解欧几里得空间中$(k,z)$-聚类问题所能达到的最优核心集尺寸。尽管该问题已取得显著进展，但现有最优上界与下界之间仍存在差距。例如，已知$k$-均值聚类（$z=2$）的最优上界为$\min \{O(k^{3/2} \varepsilon^{-2}), O(k \varepsilon^{-4})\}$ [1,2]，而最优下界为$\Omega(k\varepsilon^{-2})$ [1]。本文在上下界两方面均取得了重要进展。对于大范围参数（即$\varepsilon, k$），我们完整刻画了最优核心集尺寸。具体成果如下：（1）当$\varepsilon \geq \Omega(k^{-1/(z+2)})$时，我们提出了欧几里得$(k,z)$-聚类的新核心集下界$\Omega(k \varepsilon^{-z-2})$。结合先前的上界$\tilde{O}_z(k \varepsilon^{-z-2})$ [1]，该下界是最优的。这一新下界令人惊讶，因为近期部分工作（参见[1,2]）曾“猜想”$\Omega(k\varepsilon^{-2})$ [1]为正确下界。（2）在上界方面，我们为欧几里得$(k,z)$-聚类提供了高效的核心集构造算法，改进了核心集尺寸。特别地，针对欧几里得空间中所有$z\geq 1$的$(k,z)$-聚类问题，我们通过统一分析给出了尺寸为$\tilde{O}_z(k^{\frac{2z+2}{z+2}} \varepsilon^{-2})$的核心集。 [1] Cohen-Addad, Larsen, Saulpic, Schwiegelshohn. STOC'22. [2] Cohen-Addad, Larsen, Saulpic, Schwiegelshohn, Sheikh-Omar, NeurIPS'22.