Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios
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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.
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