An improved algorithm for computing Steiner minimal trees in Euclidean d-space

Authors:
Marcia Fampa;Kurt M. Anstreicher
Affiliations:
Institute of Mathematics, Department of Computer Sciences, Federal University of Rio de Janeiro, CP 68.530, 21941-590 Rio de Janeiro, RJ, Brazil;Department of Management Sciences, University of Iowa, Iowa City, IA 52242, USA
Venue:
Discrete Optimization
Year:
2008

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Abstract

We describe improvements to Smith's branch-and-bound (B&B) algorithm for the Euclidean Steiner problem in R^d. 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.