Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

• Published in: Discrete & computational geometry Vol. 67; no. 1; pp. 311 - 327
• Main Author: Biniaz, Ahmad
• Format: Journal Article
• Language: English
• Published: New York Springer US 2022; Springer Nature B.V
• Subjects: Algorithms; Approximation; Combinatorics; Computational Mathematics and Numerical Analysis; Euclidean geometry; Geometry; Graph theory; Graphs; Hypotheses; Mathematics; Mathematics and Statistics; Wireless networks; Approximation algorithms; Bottleneck spanning tree; Bounded-degree spanning tree; Spanning tree ratios
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-021-00286-4

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
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. (25th Annual ACM Symposium on Theory of Computing, pp. 438–447. ACM, New York (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.