Speeding up the Dreyfus–Wagner algorithm for minimum Steiner trees

Abstract

The Dreyfus–Wagner algorithm is a well-known dynamic programming method for computing minimum Steiner trees in general weighted graphs in time O ^*(3^k), where k is the number of terminal nodes to be connected. We improve its running time to O ^*(2.684^k) by showing that the optimum Steiner tree T can be partitioned into T = T ₁∪ T ₂ ∪ T ₃ in a certain way such that each T _i is a minimum Steiner tree in a suitable contracted graph G _i with less than ${\frac{k}{2}}$ terminals. In the rectilinear case, there exists a variant of the dynamic programming method that runs in O ^*(2.386^k). In this case, our splitting technique yields an improvement to O ^*(2.335^k).