Separated Red Blue Center Clustering

TL;DR

This paper introduces a generalized $k$-center clustering problem with two types of centers, red and blue, separated by a minimum distance, and provides approximation and polynomial algorithms for it.

Contribution

It extends $k$-center clustering to a two-center type with separation constraints and offers new algorithms for the generalized and line-constrained versions.

Findings

Developed an approximation algorithm for the generalized problem.

Designed a polynomial-time algorithm for the line-constrained case.

Achieved efficient solutions for clustering with separation constraints.

Abstract

We study a generalization of $k$-center clustering, first introduced by Kavand et. al., where instead of one set of centers, we have two types of centers, $p$ red and $q$ blue, and where each red center is at least $\alpha$ distant from each blue center. The goal is to minimize the covering radius. We provide an approximation algorithm for this problem, and a polynomial time algorithm for the constrained problem, where all the centers must lie on a line $\ell$.