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Designing small-sized \emph{coresets}, which approximately preserve the costsof the solutions for large datasets, has been an important research directionfor the past decade. We consider coreset construction for a variety of generalconstrained clustering problems. We introduce a general class of assignmentconstraints, including capacity constraints on cluster centers, and assignmentstructure constraints for data points (modeled by a convex body $\mathcal{B}$).We give coresets for clustering problems with such general assignmentconstraints that significantly generalizes and improves known results. Notableimplications include the first $\epsilon$-coreset for capacitated and fair$k$-Median with $m$ outliers in Euclidean spaces whose size is $\tilde{O}(m +k^2 \epsilon^{-4})$, generalizing and improving upon the prior bounds in[Braverman et al., FOCS' 22; Huang et al., ICLR' 23] (for capacitated$k$-Median, the coreset size bound obtained in [Braverman et al., FOCS' 22] is$\tilde{O}(k^3 \epsilon^{-6})$, and for $k$-Median with $m$ outliers, thecoreset size bound obtained in [Huang et al., ICLR' 23]} is $\tilde{O}(m + k^3\epsilon^{-5})$), and the first $\epsilon$-coreset of size $\mathrm{poly}(k\epsilon^{-1})$ for fault-tolerant clustering for various types of metricspaces.

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