Tight Coreset Bounds for Euclidean $(k,z)$-Clustering
We study the optimal coreset size for Euclidean $(k,z)$-clustering, where $z\ge1$ is fixed. Huang, Li, and Wu (STOC 2024) proved a lower bound of $Ω_z(k\varepsilon^{-z-2})$ for $\varepsilon\gtrsim_z k^{-1/(z+2)}$, matching the known upper bound up to polylogarithmic factors. For $0<\varepsilon<k^{-1/(z+2)}$, the upper bound $\widetilde O_z(k^{(2z+2)/(z+2)}\varepsilon^{-2})$ remained unmatched. The previous lower bounds gave only $Ω_z(\max\{k\varepsilon^{-2},k^{(2z+3)/(z+2)}\varepsilon^{-1}\})$. We close this gap by proving a matching lower bound throughout $0<\varepsilon<k^{-1/(z+2)}$, including arbitrarily small $\varepsilon$. Together with known bounds, this determines the optimal worst-case coreset size as $$ \widetildeΘ_z\!\left(\min\left\{k^{(2z+2)/(z+2)}\varepsilon^{-2},\,k\varepsilon^{-z-2}\right\}\right) $$ for every integer $k\ge1$ and $0<\varepsilon<1$. In particular, the bounds are $\widetildeΘ(k^{4/3}\varepsilon^{-2})$ for $k$-median when $\varepsilon<k^{-1/3}$, and $\widetildeΘ(k^{3/2}\varepsilon^{-2})$ for $k$-means when $\varepsilon<k^{-1/4}$. Our construction uses projection levels to encode four-wise independent signs, allowing $k$ centers to detect omissions regardless of how the retained points are reweighted.