Min-degree constrained minimum spanning tree problem: complexity, properties, and formulations

Given an undirected graph G = (V,E) and a function d : V → N , the Min-Degree Constrained Minimum Spanning Tree (md-MST) problem is to find a minimum cost spanning tree T of G where each node i ∈ V has minimum degree d(i) or is a leaf node. This problem is closely related with the well-known Degree Constrained Minimum Spanning Tree (d-MST) problem, where the degree constraint is an upper limit instead. In this paper we prove that the md-MST problem is NP-hard and present some proprieties, namely upper and lower limits to the number of central nodes and leaf-nodes in any feasible solution to the problem. Flow based formulations are also proposed and computational experiments involving the associated LP relaxations are presented. These results indicate that, for similar formulations to both d-MST and md-MST problems, the LP versions of the d-MST stronger flow models seem to provide a better approximation to the integer polyhedron than the correspondent md-MST flow formulations (within the linear relaxation context), which seems to indicate that it might be harder to get good formulations to the later problem.