Approximation algorithms for $k$-median with lower-bound constraints

We study a variant of the classical $k$-median problem known as diversity-aware $k$-median (introduced by Thejaswi et al. 2021), where we are given a collection of facility subsets, and a solution must contain at least a specified number of facilities from each subset.We investigate the fixed-parameter tractability of this problem and show several negative hardness and inapproximability results, even when we afford exponential running time with respect to some parameters of the problem. Motivated by these results we present a fixed parameter approximation algorithm with approximation ratio $(1 + \frac{2}{e} +\epsilon)$, and argue that this ratio is essentially tight assuming the gap-exponential time hypothesis. We also present a simple, practical local-search algorithm that gives a bicriteria $(2k, 3+\epsilon)$ approximation with better running time bounds.