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In this work, we study a range of constrained versions of the $k$-supplierand $k$-center problems such as: capacitated, fault-tolerant, fair, etc. Theseproblems fall under a broad framework of constrained clustering. A unifiedframework for constrained clustering was proposed by Ding and Xu [SODA 2015] incontext of the $k$-median and $k$-means objectives. In this work, we extendthis framework to the $k$-supplier and $k$-center objectives. This unifiedframework allows us to obtain results simultaneously for the followingconstrained versions of the $k$-supplier problem: $r$-gather, $r$-capacity,balanced, chromatic, fault-tolerant, strongly private, $\ell$-diversity, andfair $k$-supplier problems, with and without outliers. We obtain the followingresults: We give $3$ and $2$ approximation algorithms for the constrained$k$-supplier and $k$-center problems, respectively, with $\mathsf{FPT}$ runningtime $k^{O(k)} \cdot n^{O(1)}$, where $n = |C \cup L|$. Moreover, theseapproximation guarantees are tight; that is, for any constant $\epsilon>0$, noalgorithm can achieve $(3-\epsilon)$ and $(2-\epsilon)$ approximationguarantees for the constrained $k$-supplier and $k$-center problems in$\mathsf{FPT}$ time, assuming $\mathsf{FPT} \neq \mathsf{W}[2]$. Furthermore,we study these constrained problems in outlier setting. Our algorithm gives $3$and $2$ approximation guarantees for the constrained outlier $k$-supplier and$k$-center problems, respectively, with $\mathsf{FPT}$ running time$(k+m)^{O(k)} \cdot n^{O(1)}$, where $n = |C \cup L|$ and $m$ is the number ofoutliers.

Tight FPT Approximation for Constrained k-Center and k-Supplier