Non-Uniform $k$-Center and Greedy Clustering

Abstract

In the Non-Uniform k-Center (NUkC) 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-NUkC, 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-NUkC is not possible if t is unbounded. 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-NUkC 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-NUkC is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [To appear in 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 kcenter 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-NUkC problem, eventually bringing us closer to the resolution of the CGK conjecture.

Techniques for Generalized Colorful $k$-Center Problems

“…It is inspired by recent insights of [IV21] on decoupling multiple covering constraints and iteratively applying a greedy partitioning procedure of [CKMN01]. By taking into account multiple colors at the same time, our framework gives an improved way of dealing with multiple covering constraints while also becoming more versatile.…”

“…A careful construction and analysis (which takes into account the earlier colors) then shows that this yields a (2L + 10, r)-partition of the γ-colorful space. Our refined charging scheme improves on a decoupled analysis of [IV21] (which gives an O(5 γ ) approximation algorithm for γ-Colorful k-Center).…”

“…While the robust setting is amenable to a variety of well-known and basic algorithmic techniques, the only constant-factor approximations for the colorful setting, which imposes multiple covering constraints leading to more balanced clusterings, are based on significantly more sophisticated techniques, tailored specifically to those settings. More precisely, three distinct techniques have been successful at achieving constant-factor approximations in the context of colorful k-center clustering, namely the combinatorial approach of [JSS21], the round-or-cut-based approach of [AAKZ21], and the iterative greedy reductions of [IV21].…”