Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

TL;DR

This paper investigates the ratios of bottleneck edge lengths in Euclidean bounded-degree spanning trees, providing improved bounds and resulting in better approximation algorithms for degree-3 and degree-4 cases.

Contribution

The paper introduces new bounds for the ratios of bottleneck spanning trees with degree constraints, enhancing approximation algorithms and revealing structural properties of Euclidean MSTs.

Findings

Improved bounds: 1; eta_2 \u2265 7, eta_3 3, eta_4 2.

Enhanced approximation algorithms for degree-3 and degree-4 bottleneck spanning trees.

Structural properties of Euclidean minimum spanning trees of independent interest.

Abstract

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 $\beta_K$ 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 $\beta_2\ge 2$, $\beta_3\ge \sqrt{2}$, and $\beta_4>1.175$. It is implied by the Hamiltonicity of the cube of the bottleneck spanning tree that $\beta_2\le 3$. The degree-3 spanning tree algorithm of Ravi et al. (STOC 1993) implies that $\beta_3\le 2$. Andersen and Ras (Networks, 68(4):302-314, 2016) showed that $\beta_4\le \sqrt{3}$. We present the following improved bounds: $\beta_2\ge\sqrt{7}$, $\beta_3\le \sqrt{3}$, 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.