Balanced $k$-Center Clustering When $k$ Is A Constant

TL;DR

This paper presents a nearly linear time 4-approximation algorithm for balanced k-center clustering with size constraints, improving approximation ratio and efficiency over previous methods, applicable to any metric space.

Contribution

Introduces a simple, efficient algorithm for balanced k-center clustering with size bounds, achieving better approximation and faster runtime for constant k.

Findings

Achieves a 4-approximation ratio.

Runs in nearly linear time.

Extends to any metric space.

Abstract

The problem of constrained $k$-center clustering has attracted significant attention in the past decades. In this paper, we study balanced $k$-center cluster where the size of each cluster is constrained by the given lower and upper bounds. The problem is motivated by the applications in processing and analyzing large-scale data in high dimension. We provide a simple nearly linear time $4$-approximation algorithm when the number of clusters $k$ is assumed to be a constant. Comparing with existing method, our algorithm improves the approximation ratio and significantly reduces the time complexity. Moreover, our result can be easily extended to any metric space.