A branch-and-cut algorithm for the minimum branch vertices spanning tree problem.

We model the MBVP as an integer linear program with undirected variables.We derive valid inequalities for it and prove that some of these are facet defining.We develop a hybrid formulation containing undirected and directed variables.We solve both models by branch-and-cut.We report computational results showing the superiority of the hybrid formulation. Given a connected undirected graph G = ( V , E ) , the Minimum Branch Vertices Problem (MBVP) asks for a spanning tree of G with the minimum number of vertices having degree greater than two in the tree. These are called branch vertices. This problem, with applications in the context of optical networks, is known to be NP-hard. We model the MBVP as an integer linear program, with undirected variables, we derive valid inequalities and we prove that some of these are facet defining. We then develop a hybrid formulation containing undirected and directed variables. Both models are solved with branch-and-cut. Comparative computational results show the superiority of the hybrid formulation.