Degree-bounded minimum spanning trees

Abstract

Given $n$ points in the Euclidean plane, the degree-$\delta$ minimum spanning tree (MST) problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most $\delta$. The problem is NP-hard for $2\le \delta \le 3$, while the NP-hardness of the problem is open for $\delta =4$. The problem is polynomial-time solvable when $\delta =5$. 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 $(\sqrt{2}+2)/3<1.1381$ times the weight of an MST.