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

Abstract

This paper addresses a new combinatorial problem, the Min‐Degree Constrained Minimum Spanning Tree (md‐MST), that can be stated as: given a weighted undirected graph with positive costs on the edges and a node‐degrees function , the md‐MST is to find a minimum cost spanning tree T of G, where each node i of T either has at least a degree of or is a leaf node. This problem is closely related to the well‐known Degree Constrained Minimum Spanning Tree (d‐MST) problem, where the degree constraint is an upper bound instead.

The general NP‐hardness for the md‐MST is established and some properties related to the feasibility of the solutions for this problem are presented, in particular we prove some bounds on the number of internal and leaf nodes. Flow‐based formulations are proposed and computational experiments involving the associated Linear Programming (LP) relaxations are presented.