In this work, we study a range of constrained versions of the k-supplier and k-center problems. In the classical (unconstrained) k -supplier problem, we are given a set of clients C in a metric space X , with distance function d (. ,.). We are also given a set of feasible facility locations L ⊆ X. The goal is to open a set F of k facilities in L to minimize the maximum distance of any client to the closest open facility, i.e., minimize, cost (F , C) ≡ max j ∈ C { d (F , j) } , where d (F , j) is the distance of client j to the closest facility in F. The k -center problem is a special case of the k -supplier problem where L = C. We study various constrained versions of the k -supplier problem such as: capacitated , fault-tolerant , ℓ -diversity, etc. These problems fall under a broad framework of constrained clustering. A unified framework for constrained clustering was proposed by Ding and Xu [Algorithmica 2020] in the context of the k -median and k -means objectives. We extend this framework to the k -supplier and k -center objectives in this work. 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 design Fixed-Parameter Tractable (FPT) algorithms for these problems. FPT algorithms have polynomial running time if the parameter under consideration is a constant. This may be relevant even to a practitioner since the parameter k is a small number in many real clustering scenarios. We obtain the following results: • We give 3 and 2 approximation algorithms for the constrained k -supplier and k -center problems, respectively, with FPT running time k O (k) ⋅ n O (1) , where n = | C ∪ 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 FPT time, assuming FPT ≠ W [ 2 ]. • We study the constrained clustering problem with outliers. Our algorithm gives 3 and 2 approximation guarantees for the constrained outlier k -supplier and k -center problems, respectively, with FPT running time (k + m) O (k) ⋅ n O (1) , where n = | C ∪ L | and m is the number of outliers. • Our techniques generalise for distance function d (. ,.) z. That is, for any positive real number z , if the cost of a client is defined by d (. ,.) z instead of d (. ,.) , then our algorithm gives 3 z and 2 z approximation guarantees for the constrained k -supplier and k -center problems, respectively. • We design 3 approximation algorithm for a wide range of constrained k -supplier problems with FPT running time f (k). p o l y (n). • We design 2 approximation algorithm for a range of constrained k -center problems with FPT running time f (k). p o l y (n). • We extend the algorithm to outlier setting with FPT running time f (k , m). p o l y (n) , where m = number of outliers. • We show matching lower bounds for the constrained k -supplier and k -center problems in the outlier and non-outlier settings. [ABSTRACT FROM AUTHOR]