Constraint Programming for the Diameter Constrained Minimum Spanning Tree Problem

Given an undirected connected graph G = (V,E) with a set V of vertices, a set E of edges, and costs cij associated to every edge [i, j] ∈ E, with i < j, the Diameter Minimum Spanning Tree Problem (DCMST) consists in finding a minimum spanning tree T = (V,E ), with E ′ ⊆ E, where the diameter required does not exceed a given positive integer value D, where 2 ≤ D ≤ |V | − 1. The diameter of a tree T is equal to the number of edges in the longest path between any two nodes i, j ∈ V in T . This problem is NP hard when D ≥ 4. DCMST applications appear in telecommunications, data compression and distributed mutual exclusion in parallel computing. Some mixed integer programming (MIP) formulations for DCMST implicitly use a property [6] that ensures that a central vertex c ∈ V exists in any feasible tree T when D is even, such that no vertex is more than D/2 edges away from c. If D is odd, a central edge e = [a, b] ∈ E exists in T , such that no vertex is more than (D − 1)/2 edges away from the closest extremity of e. The first DCMST formulations, using single commodity flows, were presented in [1,2]. Improved formulations with valid inequalities and a lifting procedure are found in [2,8]. An alternative formulation for the odd D case