A Bottleneck Matching Problem with Edge-Crossing Constraints

Motivated by a crane assignment problem, we consider a Euclidean bipartite matching problem with edge-crossing constraints. Speci cally, given n red points and n blue points in the plane, we want to construct a perfect matching between red and blue points that minimizes the length of the longest edge, while imposing a constraint that no two edges may cross each other. We show that the problem is NP-hard and give an algorithm that solves our problem in O(n log n) time when the red and blue points form the vertices of a convex polygon.