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 $\max{r_1,r_2}$ is minimized, where $r_1$ (resp., $r_2$) is the radius of the smallest enclosing disk of all red (resp., blue) points. Previously, an exact algorithm of $O(n^3\log^2 n)$ time and a $(1+\varepsilon)$-approximate algorithm of $O(n + (1/\varepsilon)^6 \log^2 (1/\varepsilon))$ time were known. In this paper, we propose a new exact algorithm of $O(n^2\log^2 n)$ time and a new $(1+\varepsilon)$-approximate algorithm of $O(n + (1/\varepsilon)^3 \log^2 (1/\varepsilon))$ time.