Min-degree constrained minimum spanning tree problem with fixed centrals and terminals: Complexity, properties and formulations

We consider a variant of the Min-Degree Constrained Minimum Spanning Tree Problem where the central and terminal nodes are fixed a priori. We prove that the optimization problem is NP-Hard even for complete graphs and the feasibility problem is NP-Complete even if there is an edge between each central and each terminal in the input graph. Actually, this complexity result still holds when the minimum degree of each central node is restricted to be a same value d ≥ 2. We derive necessary and sufficient conditions for feasibility. We present several integer linear programming formulations - based on known formulations for the minimum spanning tree problem - along with a theoretical comparison among the lower bounds provided by their linear relaxations. We propose three Lagrangian heuristics. Computational experiments compare the performances of the heuristics and the formulations.