Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

Abstract

Inspired by the seminal works of Khuller et al. (SIAM J. Comput. 25(2), 355–368 (1996)) and Chan (Discrete Comput. Geom. 32(2), 177–194 (2004)) we study the bottleneck version of the Euclidean bounded-degree spanning tree problem. A bottleneck spanning tree is a spanning tree whose largest edge-length is minimum, and a bottleneck degree-K spanning tree is a degree-K spanning tree whose largest edge-length is minimum. Let $\beta _K$ be the supremum ratio of the largest edge-length of the bottleneck degree-K spanning tree to the largest edge-length of the bottleneck spanning tree, over all finite point sets in the Euclidean plane. It is known that $\beta _5=1$, and it is easy to verify that $\beta _2\geqslant 2$, $\beta _3\geqslant \sqrt{2}$, and $\beta _4>1.175$. It is implied by the Hamiltonicity of the cube of the bottleneck spanning tree that $\beta _2\leqslant 3$. The degree-3 spanning tree algorithm of Ravi et al. (25th Annual ACM Symposium on Theory of Computing, pp. 438–447. ACM, New York (1993)) implies that $\beta _3\leqslant 2$. Andersen and Ras (Networks 68(4), 302–314 (2016)) showed that $\beta _4\leqslant \sqrt{3}$. We present the following improved bounds: $\beta _2\geqslant \sqrt{7}$, $\beta _3\leqslant \sqrt{3}$, and $\beta _4\leqslant \sqrt{2}$. As a result, we obtain better approximation algorithms for Euclidean bottleneck degree-3 and degree-4 spanning trees. As parts of our proofs of these bounds we present some structural properties of the Euclidean minimum spanning tree which are of independent interest.