Minimizing the Number of Separating Circles for Two Sets of Points in the Plane

Abstract

Given two sets of points $\mathbb{R}$ and $\mathbb{B}$ in the plane, we address the problem of finding a set of circles $\mathbb{C} = \{c_i, i= 1, 2, \ldots, k\}$, satisfying the condition that every point in $\mathbb{R}$ is covered by at least one circle $c_i$ and each point in $\mathbb{B}$ is not covered by any circle $c_i, i = 1, 2, \ldots k$. We conjecture that to find such a set with the smallest $k$ is NP-complete. In this paper, we present an approximation algorithm for computing the set with minimal number of such circles. The algorithm finds also a lower bound of the smallest $k$.