Geodesic
The Minimum Moving Spanning Tree Problem
Journal of Graph Algorithms and Applications, Tome 27 (2023) no. 1, pp. 1-18.

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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 $\varepsilon>0$.
DOI : 10.7155/jgaa.00607
Keywords: minimum spanning tree, moving points, NP-hardness, convex distance function, approximation algorithms 
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Hugo Akitaya; Ahmad Biniaz; Prosenjit Bose; Jean-Lou De Carufel; Anil Maheshwari; Luís Fernando Schultz Xavier da Silveira; Michiel Smid. The Minimum Moving Spanning Tree Problem. Journal of Graph Algorithms and Applications, Tome 27 (2023) no. 1, pp. 1-18. doi : 10.7155/jgaa.00607. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00607/

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