Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

Inspired by the seminal works of Khuller et al. (STOC 1994) and Chan (SoCG 2003) 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 $\betaK$ 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 $\beta5=1$, and it is easy to verify that $\beta2\ge 2$, $\beta3\ge \sqrt{2}$, and $\beta4>1.175$. It is implied by the Hamiltonicity of the cube of the bottleneck spanning tree that $\beta2\le 3$. The degree-3 spanning tree algorithm of Ravi et al. (STOC 1993) implies that $\beta3\le 2$. Andersen and Ras (Networks, 68(4):302-314, 2016) showed that $\beta4\le \sqrt{3}$. We present the following improved bounds: $\beta2\ge\sqrt{7}$, $\beta3\le \sqrt{3}$, and $\beta_4\le \sqrt{2}$. 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.