Degree-bounded minimum spanning trees

Abstract

Given n points in the Euclidean plane, the degree- δ minimum spanning tree (MST) problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most δ . The problem is NP-hard for 2 ≤ δ ≤ 3 , while the NP-hardness of the problem is open for δ = 4 . The problem is polynomial-time solvable when δ = 5 . By presenting an improved approximation analysis for Chan’s degree-4 MST algorithm [T. Chan, Euclidean bounded-degree spanning tree ratios, Discrete & Computational Geometry 32 (2004) 177–194], we show that, for any arbitrary collection of points in the Euclidean plane, there always exists a degree-4 spanning tree of weight at most ( 2 + 2 ) / 3 < 1.1381 times the weight of an MST.