Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

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 β 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 β 5 = 1, and it is easy to verify that β 2 2, β 3 √ 2, and β 4 > 1.175. It is implied by the Hamiltonicity of the cube of the bottleneck spanning tree that β 2 3. The degree-3 spanning tree algorithm of Ravi et al. (STOC 1993) implies that β 3 2. Andersen and Ras (Networks, 68(4):302-314, 2016) showed that β 4 √ 3. We present the following improved bounds: β 2 √ 7, β 3 √ 3, and β 4 √ 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. * Part of this work has been done while the author was an NSERC postdoctoral fellow at University of Waterloo. 1 The cube of a graph G has the same vertices as G, and has an edge between two distinct vertices if and only if there exists a path, with at most three edges, between them in G.