Non-Uniform k-Center and Greedy Clustering

In the Non-Uniform k -Center (NU k C) problem, a generalization of the famous k -center clustering problem, we want to cover the given set of points in a metric space by finding a placement of balls with specified radii. In t -NU k C, we assume that the number of distinct radii is equal to t , and we are allowed to use k i balls of radius r i , for 1 ≤ i ≤ t . This problem was introduced by Chakrabarty et al. [ACM Trans. Alg. 16(4):46:1-46:19], who showed that a constant approximation for t -NU k C is not possible if t is unbounded, assuming P ̸ = NP . On the other hand, they gave a bicriteria approximation that violates the number of allowed balls as well as the given radii by a constant factor. They also conjectured that a constant approximation for t -NU k C should be possible if t is a fixed constant. Since then, there has been steady progress towards resolving this conjecture – currently, a constant approximation for 3-NU k C is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [SOSA 2022]. We push the horizon by giving an O (1)-approximation for the Non-Uniform k -Center for 4 distinct types of radii. Our result is obtained via a novel combination of tools and techniques from the k -center literature, which also demonstrates that the different generalizations of k -center involving non-uniform radii, and multiple coverage constraints (i.e., colorful k -center ), are closely interlinked with each other. We hope that our ideas will contribute towards a deeper understanding of the t -NU k C problem, eventually bringing us closer to the resolution of the CGK conjecture.