Red blue $k$-center clustering with distance 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 $\ell$.