A Hybrid Relax-and-Cut/Branch and Cut Algorithm for the Degree-Constrained Minimum Spanning Tree Problem

A new exact solution algorithm is proposed for the Degree-Constrained Minimum Spanning Tree Problem. The algorithm involves two combined phases. The first one contains a Lagrangian Relaxand-Cut procedure while the second implements a Branch-and-Cut algorithm. Both phases rely on a standard formulation for the problem, reinforced with Blossom Inequalities. An important feature of the proposed algorithm is that, whenever optimality is not proven by Relax-and-Cut alone, an attractive set of valid inequalities is identified and carried over, in a warm start, from Lagrangian Relaxation to Branch and Cut. In doing so, without the need of running any separation algorithm, one aims at obtaining an initial Linear Programming relaxation bound at least as good as the best Lagrangian bound previously obtained. Computational results indicate that our hybrid algorithm dominates similar hybrid algorithms based on the standard formulation alone. Likewise, it also dominates pure Branch and Cut algorithms based on either the standard formulation or its Blossom Inequalities reinforced version.