A new heuristic for the Euclidean Steiner Tree Problem in ${\mathbb {R}}^n$

Abstract

Given p points in ${\mathbb {R}}^n$, called terminal points, the Euclidean Steiner Tree Problem (ESTP) consists of finding the shortest tree connecting them, using or not extra points, called Steiner points. This is a well-known NP-hard combinatorial optimization problem. Instances with thousands of points have been solved for $n=2$. However, methods specialized for the ESTP in ${\mathbb {R}}^2$ cannot be applied to problems in higher dimensions. Enumeration schemes have been proposed in the literature. Unfortunately, the number of Steiner trees having p terminal points grows extremely fast with p, so the enumeration of all trees is only possible for very small values of p. For $n\ge 3$, even small instances with tens of points cannot be solved with exact algorithms in a reasonable time. In this work, we present two heuristics for the ESTP. These heuristics differ from most existent ones in the literature in the fact that they do not rely on the minimum spanning tree of the terminal points. Instead, they start with a single extra point connected to all terminal points and new extra points are introduced iteratively according to angle properties for two consecutive edges. The heuristics return the optimal solution in most of the small test instances. For large instances, where the optimum is not known, the heuristics return relatively good solutions, according to their Steiner ratio.