Covering a bichromatic point set with two disjoint monochromatic disks

Let P be a set of n points in the plane in general position such that its elements are colored red or blue. We study the problem of finding two disjoint disks D-r and D-b such that D-r covers only red points, D-b covers only blue points, and the number of elements of P contained in D-r U D-b is maximized. We prove that this problem can be solved in O(n(11/3) polylog n) time. We also present a randomized algorithm that with high probability returns a (1 - epsilon)-approximation to the optimal solution in O(n(4/3) epsilon(-6) polylogn) time. (C) 2012 Elsevier B.V. All rights reserved.