Non-Uniform k-Center and Greedy Clustering

by   Tanmay Inamdar, et al.

In the Non-Uniform k-Center 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 Problem, 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 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-NUkC problem, eventually bringing us closer to the resolution of the CGK conjecture.


page 1

page 2

page 3

page 4


Towards Non-Uniform k-Center with Constant Types of Radii

In the Non-Uniform k-Center problem we need to cover a finite metric spa...

On Perturbation Resilience of Non-Uniform k-Center

The Non-Uniform k-center (NUkC) problem has recently been formulated by ...

Robust k-Center with Two Types of Radii

In the non-uniform k-center problem, the objective is to cover points in...

A Constant Approximation for Colorful k-Center

In this paper, we consider the colorful k-center problem, which is a gen...

A dichotomy theorem for nonuniform CSPs simplified

In a non-uniform Constraint Satisfaction problem CSP(G), where G is a se...

Techniques for Generalized Colorful k-Center Problems

Fair clustering enjoyed a surge of interest recently. One appealing way ...

Fair Colorful k-Center Clustering

An instance of colorful k-center consists of points in a metric space th...

Please sign up or login with your details

Forgot password? Click here to reset