Improved Coresets for Clustering with Capacity and Fairness Constraints

We study coresets for clustering with capacity and fairness constraints. Our main result is a near-linear time algorithm to construct $\tilde{O}(k^{2\varepsilon{-2z-2})$-sized} $\varepsilon$-coresets for capacitated $(k,z)$-clustering which improves a recent $\tilde{O}(k^{3\varepsilon{-3z-2})$} bound by [BCAJ+22, HJLW23]. As a corollary, we also save a factor of $k \varepsilon^{-z}$ on the coreset size for fair $(k,z)$-clustering compared to them. We fundamentally improve the hierarchical uniform sampling framework of [BCAJ+22] by adaptively selecting sample size on each ring instance, proportional to its clustering cost to an optimal solution. Our analysis relies on a key geometric observation that reduces the number of total effective centers" from [BCAJ+22]'s $\tilde{O}(k^2\varepsilon^{-z})$ to merely $O(k\log \varepsilon^{-1})$ by being able toignore'' all center points that are too far or too close to the ring center.