Planar bichromatic bottleneck spanning trees

Abstract

Given a set $P$ of red and blue points in the plane, a planar bichromatic spanning tree of $P$ is a geometric spanning tree of $P$, such that each edge connects a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree $T$, such that the length of the longest edge (i.e., bottleneck) in $T$ is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time $(8\sqrt{2})$-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck $\lambda$ can be converted to a planar bichromatic spanning tree of bottleneck at most $8\sqrt{2}\lambda$.