Binary Steiner trees: Structural results and an exact solution approach

In this paper we study the Steiner tree problem with degree constraints. Motivated by an application in computational biology we focus on binary Steiner trees in which all node degrees are required to be at most three. It is shown that finding a binary Steiner tree is NPNP-complete for arbitrary graphs. We relate the problem to Steiner trees without degree constraints as well as degree-constrained spanning trees by proving approximation ratios. Further, we give integer programming formulations for this problem on undirected and directed graphs and study the associated polytopes for both cases. Some classes of facets are introduced. Based on this study a branch-&-cut approach is developed and evaluated on biological instances coming from the reconstruction of phylogenetic trees. We are able to solve nearly all instances up to 200 nodes to optimality within a limited amount of time. This shows the effectiveness of our approach.