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

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.

A new second‐order conic optimization model for the Euclidean Steiner tree problem in Rd$\mathbb {R}^d$

“…The first enumeration‐based approaches are the two algorithms proposed in Gilbert and Pollak (1968) and Smith (1992). For more detail, see Fampa and Anstreicher (2008), Fonseca et al. (2014), D'Ambrosio et al.…”