Tight Coreset Bounds for $(k,z)$-Clustering in Euclidean and Doubling Metrics
We determine the optimal worst-case size of strong coresets for $(k,z)$-clustering in Euclidean spaces and doubling metrics, up to polylogarithmic factors. In Euclidean space, previous lower bounds matched the best upper bounds only for $\varepsilon\gtrsim k^{-1/(z+2)}$. We close the gap for arbitrarily small $\varepsilon$, establishing the optimal size $\widetildeΘ(\min\{k^{(2z+2)/(z+2)}\varepsilon^{-2},\,k\varepsilon^{-z-2}\})$ for every fixed $z\ge1$, all $k\ge1$, and $0<\varepsilon<1$. Our lower bound replaces each offset in a previous hard instance by a group of distinct offsets. A fixed family of $k$-center queries and four-wise independent signs force a large relative cost error for every sufficiently small subset with arbitrary nonnegative weights. For doubling metrics of dimension $D$, we resolve the previously unmatched bounds for $z>2$. We prove an upper bound of $\widetilde O(k^{2-2/z}D\varepsilon^{-2})$ and a lower bound that, together with known upper bounds, establishes the optimal size $\widetildeΘ(kD\min\{k^{1-2/z}\varepsilon^{-2},\,\varepsilon^{-z}\})$ for fixed $z>2$, $0<\varepsilon\le\varepsilon_z$, and $D\ge C_z\log(2/\varepsilon)$, where $C_z>0$ and $\varepsilon_z\in(0,1)$ depend only on $z$. The lower bound exploits the ability to encode $Θ(D)$ independent signs per center without diluting relative cost discrepancy. For the upper bound, we use independent, unbiased two-point summaries that exactly preserve cluster mass and anchor cost, combined with iterative size reduction.