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

In this work, we study a range of constrained versions of the k-supplier and k-center problems such as: capacitated, fault-tolerant, fair, etc. These problems fall under a broad framework of constrained clustering. A unified framework for constrained clustering was proposed by Ding and Xu [SODA 2015] in context of the k-median and k-means objectives. In this work, we extend this framework to the k-supplier and k-center objectives. This unified framework allows us to obtain results simultaneously for the following constrained versions of the k-supplier problem: r-gather, r-capacity, balanced, chromatic, fault-tolerant, strongly private, ℓ-diversity, and fair k-supplier problems, with and without outliers. We obtain the following results: We give 3 and 2 approximation algorithms for the constrained k-supplier and k-center problems, respectively, with mathsfFPT running time kO(k) · nO(1), where n = |C cup L|. Moreover, these approximation guarantees are tight; that is, for any constant ε>0, no algorithm can achieve (3-ε) and (2-ε) approximation guarantees for the constrained k-supplier and k-center problems in mathsfFPT time, assuming mathsfFPT neq mathsfW[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 mathsfFPT running time (k+m)O(k) · nO(1), where n = |C cup L| and m is the number of outliers.