Improved algorithms for the bichromatic two-center problem for pairs of points

Abstract

We consider a bichromatic two-center problem for pairs of points. Given a set S of n pairs of points in the plane, for every pair, we want to assign a red color to one point and a blue color to the other, in such a way that the value max { r1, r2} is minimized, where r1 (resp., r2) is the radius of the smallest enclosing disk of all red (resp., blue) points. Previously, an exact algorithm of O(n3log 2n) time and a (1 + ε) -approximate algorithm of O(n+ (1/ε) 6log 2(1/ε)) time were known. In this paper, we propose a new exact algorithm of O(n2log 2n) time and a new (1 + ε) -approximate algorithm of O(n+ (1/ε) 3log 2(1/ε)) time.