Coresets for Constrained Clustering: General Assignment Constraints and Improved Size Bounds

Designing small-sized \emph{coresets}, which approximately preserve the costs of the solutions for large datasets, has been an important research direction for the past decade. We consider coreset construction for a variety of general constrained clustering problems. We introduce a general class of assignment constraints, including capacity constraints on cluster centers, and assignment structure constraints for data points (modeled by a convex body $\mathcal{B}$). We give coresets for clustering problems with such general assignment constraints that significantly generalizes and improves known results. Notable implications include the first $\epsilon$-coreset for capacitated and fair $k$-Median with $m$ outliers in Euclidean spaces whose size is $\tilde{O}(m + k² \epsilon^{-4})$, generalizing and improving upon the prior bounds in Braverman et al., FOCS' 22; Huang et al., ICLR' 23, and the first $\epsilon$-coreset of size $\mathrm{poly}(k \epsilon^{-1})$ for fault-tolerant clustering for various types of metric spaces.