The Minimum Moving Spanning Tree Problem.

We investigate the problem of finding a spanning tree of a set of moving points in the plane that minimizes the maximum total weight (sum of Euclidean distances between edge endpoints) or the maximum bottleneck throughout the motion. The output is a single tree, i.e., it does not change combinatorially during the movement of the points. We call these trees the minimum moving spanning tree, and the minimum bottleneck moving spanning tree, respectively. We show that, although finding the minimum bottleneck moving spanning tree can be done in \(O(n^2)\) time, it is NP-hard to compute the minimum moving spanning tree. We provide a simple \(O(n^2)\)-time 2-approximation and a \(O(n \log n)\)-time \((2+\varepsilon )\)-approximation for the latter problem.