Red blue $k$-center clustering with distance constraints

TL;DR

This paper studies a constrained $k$-center clustering problem in Euclidean space, introducing algorithms for dividing centers into red and blue groups with distance constraints and optimizing placement on a line.

Contribution

It presents a bi-criteria approximation algorithm for the constrained clustering problem and a polynomial-time solution for centers on a line.

Findings

Developed a bi-criteria approximation algorithm for the red-blue $k$-center problem.

Provided a polynomial-time algorithm for centers constrained on a line.

Achieved minimized covering radius under distance and placement constraints.

Abstract

We consider a variant of the $k$-center clustering problem in $\Re^d$, where the centers can be divided into two subsets, one, the red centers of size $p$, and the other, the blue centers of size $q$, where $p+q=k$, and such that each red center and each blue center must be apart a distance of at least some given $\alpha \geq 0$, with the aim of minimizing the covering radius. We provide a bi-criteria approximation algorithm for the problem and a polynomial time algorithm for the constrained problem where all centers must lie on a given line $\ell$.