A Dynamic Adaptive Relaxation Scheme Applied to the Euclidean Steiner Minimal Tree Problem

Abstract

The Steiner problem is an NP-hard optimization problem which consists of finding the minimal-length tree connecting a set of N points in the Euclidean plane. Exact methods of resolution currently available are exponential in N, making exact minimal trees accessible for only small size problems (up to N \approx 100). An acceptable suboptimal solution is provided by the minimum spanning tree (MST) which has been shown computable in an O(N log N) step. We propose here an O(N) process that is able to relax a given initial Steiner tree into a local minimum of its length. This process, based on a physical analogy, simulates the dynamics of a fluid film which relaxes under surface tension forces and stabilizes in an equilibrium configuration minimizing its total length, through purely local interactions. To improve the solution to the Steiner problem, this O(N) relaxation scheme is applied to reduce the length of the MST. This results in a heuristic of a very low O(N log N) complexity for the Steiner problem, whose performance is shown to compare quite favorably with that of the best available heuristics. Large problem sizes up to N=10000 were successfully tackled. A characterization of the asymptotic behavior of the solution of the Steiner problem shows a stabilization to a nonvanishing positive value of the average length reduction achieved over the MST and predicts an average length for the minimal Steiner tree of about 3% below 0.65 N1/2 for large N.