The Minimum Moving Spanning Tree Problem

Abstract

We investigate the problem of finding a spanning tree of a set of $n$ moving points in $\mathbb{R}^{\dim}$ that minimizes the maximum total weight (under any convex distance function) 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 a minimum moving spanning tree, and a 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 when $\dim$ is a constant, it is NP-hard to compute the minimum moving spanning tree even for $\dim=2$. We provide a simple $O(n^2)$-time 2-approximation and a $O(n \log n)$-time $(2+\varepsilon)$-approximation for the latter problem, for any constant $\dim$ and any constant $\varepsilon>0$.