Approximation Algorithms for Diversity-Bounded Center Problems

This paper considers the diversity-bounded center problems, where we are given a set of points, each of which was colored in one of the $$\omega $$ colors, along with integers k and $$l_i$$ , $$u_i$$ for color i, the goal is to select a k-sized center set so as to minimize the maximum distance of a point to its nearest center, and at the same time, meet the requirements that the amount of selected centers with color i must be within $$[l_i, u_i]$$ for each i. The diversity-bounded clustering with one-side upper bound and lower bound requirement was considered in (Jones et al., 2020) and (Thejaswi et al., 2021), respectively. We combine the difficulties of them and propose the diversity-bounded center problems from both sides, and as the main contribution, we present 3-approximation algorithms for the red-blue as well as the multi-colored version, the complexity of which for the latter problem is parameterized by $$\omega $$ .