# Red–Blue k-Center Clustering with Distance Constraints

## Abstract

We consider a variant of the k-center clustering problem in (Formula presented.), 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, such that (Formula presented.), and each red center and each blue center must be a distance of at least some given (Formula presented.) apart. The aim is to minimize 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 ℓ. Additionally, we present a polynomial time algorithm for the case where only the orientation of the line is fixed in the plane ((Formula presented.)), although the algorithm works even in (Formula presented.) by constraining the line to lie in a plane and with a fixed orientation.

## Publication Title

Mathematics

## Recommended Citation

Eskandari, M., Khare, B., Kumar, N., & Sadeghi Bigham, B.
(2023). Red–Blue k-Center Clustering with Distance Constraints.* Mathematics**, 11* (3)
https://doi.org/10.3390/math11030748