Approximating Longest Spanning Tree with Neighborhoods

We study the following maximization problem in the Euclidean plane: Given a collection of neighborhoods (polygonal regions) in the plane, the goal is to select a point in each neighborhood so that the longest spanning tree on selected points has maximum length. It is not known whether or not this problem is NP-hard. We present an approximation algorithm with ratio 0.548 for this problem. This improves the previous best known ratio of 0.511. The presented algorithm takes linear time after computing a diameter. Even though our algorithm itself is fairly simple, its analysis is rather involved. In some part we deal with a minimization problem with multiple variables. We use a sequence of geometric transformations to reduce the number of variables and simplify the analysis

Spanning Trees in Multipartite Geometric Graphs

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 (Formula presented.). A bichromatic spanning tree is a spanning tree in the complete bipartite geometric graph with bipartition (R, B). The minimum (respectively maximum) bichromatic spanning tree problem is the problem of computing a bichromatic spanning tree of minimum (respectively maximum) total edge length. (1) We present a simple algorithm that solves the minimum bichromatic spanning tree problem in (Formula presented.) time. This algorithm can easily be extended to solve the maximum bichromatic spanning tree problem within the same time bound. It also can easily be generalized to multicolored point sets. (2) We present (Formula presented.)-time algorithms that solve the minimum and the maximum bichromatic spanning tree problems. (3) We extend the bichromatic spanning tree algorithms and solve the multicolored version of these problems in (Formula presented.) time, where k is the number of different colors (or the size of the multipartition in a complete multipartite geometric graph)