AN IMPROVED ALGORITHM FOR STEINER TREES

Two basic results for the Euclidean Steiner minimal tree problem are shown: (1) all admissible partitions of the set of nodes (which the tree should span) correspond to degenerate full Steiner topologies, and can be systematically derived from these topologies if degeneracy occurs; (2) for any given full Steiner tree topology there exists a lower bound on the value of the corresponding Steiner tree (this bound holds as an equality if the tree is full). Taken together these two results suggest a relatively efficient (although still exponential) branch and bound algorithm for the Steiner minimal tree construction. The algorithm uses the lower bound to discard some topologies after a brief check, and does away with the need to check all the possible partitions of the node set.