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Minimum cost degree constrained spanning trees with node-degree costs

The Degree constrained Minimum Spanning Tree Problem (DMSTP) consists in finding a minimal cost spanning tree satisfying the condition that every node has a degree no greater than a fixed value. Here we consider a variant where besides the edge costs, a concave cost function is associated to the degree of each node. Several integer linear programming formulations based on discretization techniques are presented and their linear programming relaxations are compared. A GRASP heuristic to obtain (hopefully good) feasible solutions for the problem is also presented together with a Path Relinking strategy. Computational results using instances with up to 100 nodes are presented. These results give an empirical assessment of the linear programming bounds of the different models as well as the ability of using these models to solve the problems by using an ILP package. The results also show that the proposed GRASP heuristic appears to perform rather well for this type of problems.