Maximum plane trees in multipartite geometric graphs

Abstract

A geometric graph is a graph whose vertices are points in the plane and whose edges are straight-line segments between the points. A plane spanning tree in a geometric graph is a spanning tree that is noncrossing. Let R and B be two disjoint sets of points in the plane where the points of R are colored red and the points of B are colored blue, and let n = |R ∪ B|. A bichromatic plane spanning tree is a plane spanning tree in the complete bipartite geometric graph with bipartition (R,B). In this paper we consider the maximum bichromatic plane spanning tree problem, which is the problem of computing a bichromatic plane spanning tree of maximum total edge length. 1. For the maximum bichromatic plane spanning tree problem, we present an approximation algorithm with ratio 1/4 that runs in O(n log n) time. 2. We also consider the multicolored version of this problem where the input points are colored with k > 2 colors. We present an approximation algorithm that computes a plane spanning tree in a complete k-partite geometric graph, and whose ratio is 1/6 if k = 3, and 1/8 if k ≥ 4. 3. We also revisit the special case of the problem where k = n, i.e., the problem of computing a maximum plane spanning tree in a complete geometric graph. For this problem, we present an approximation algorithm with ratio 0.503; this is an extension of the algorithm presented by Dumitrescu and Tóth (2010) whose ratio is 0.502.