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

Abstract: 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 generalize and improve known results. Notable implications include the first $\varepsilon$-coreset for capacitated and fair $k$-Median with $m$ outliers in Euclidean spaces whose size is $\tilde{O}(m + k² \varepsilon<sup>{-4})$, generalizing and improving upon the prior bounds in $k$-Median, the coreset size bound obtained in [Braverman et al., FOCS' 22] is $tilde{O}(k^3 varepsilon^{-6})$, and for $k$-Median with $m$ outliers, the coreset size bound obtained in [Huang et al., ICLR' 23]} is $tilde{O}(m + k^3 varepsilon^{-5})$" title="" rel="nofollow" data-turbo="false" class="assistant-link">Braverman et al., FOCS' 22; Huang et al., ICLR' 23, and the first $\epsilon$-coreset of size $\mathrm{poly}(k \varepsilon<sup>{-1})$ for fault-tolerant clustering for various types of metric spaces.