Disjunctive combinatorial branch in a subgradient tree algorithm for the DCMST problem with VNS-Lagrangian bounds

Consider an undirected graph G=(V,E) with a set of nodes V and a set of weighted edges E. The weight of an edge e∈E is denoted by ce⩾0. The degree constrained minimum spanning tree (DCMST) problem consists in finding a minimum spanning tree of G subject to maximum degree constraints dv∈N on the number of edges connected to each node v∈V. We propose a Variable Neighborhood Search (VNS)-Lagrangian heuristic that outperforms the best known heuristics in the literature for this problem. We use an exact subgradient tree (SGT) algorithm in order to prove solution optimality. The SGT algorithm uses a combinatorial relaxation to evaluate lower bounds on each relaxed solution. Thus, we propose a new branching scheme that separates an integer relaxed solution from the domain of spanning trees while generating new disjoint SGT-node partitions. We prove optimality for many benchmark instances and improve lower and upper bounds for the instances whose optimal solutions remain unknown.