We describe improvements to Smith's branch-and-bound (B&B) algorithm for the Euclidean Steiner problem in ℝd. Nodes in the B&B tree correspond to full Steiner topologies associated with a subset of the terminal nodes, and branching is accomplished by ‘merging’ a new terminal node with each edge in the current Steiner tree. For a given topology we use a conic formulation for the problem of locating the Steiner points to obtain a rigorous lower bound on the minimal tree length. We also show how to obtain lower bounds on the child problems at a given node without actually computing the minimal Steiner trees associated with the child topologies. These lower bounds reduce the number of children created and also permit the implementation of a ‘strong branching’ strategy that varies the order in which the terminal nodes are added. Computational results demonstrate substantial gains compared to Smith's original algorithm.
