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

Abstract

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 ≈ 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 0.65N 1/2 for large N .

Iterated local search algorithms for the Euclidean Steiner tree problem in n dimensions

“…The second procedure is the same procedure used for constructing an initial solution for the ESTP in two dimensions in the dynamic relaxation algorithm introduced by Chapeau‐Blondeau et al. () and improved by Montenegro et al. () for n dimension.…”

“…Figure illustrates this movement. This movement is equivalent to the one used in the interaction procedure from the dynamic relaxation method (Chapeau‐Blondeau et al., ), being used, as well, every time two Steiner points are connected by a zero‐length connection. After applying this movement to all pair of Steiner points connected by a zero‐length connection, in a relative minimum tree with full Steiner topology, we can guarantee that the resulting tree will be a Steiner tree with a smaller length than the original (Thompson, ).…”

“…The heuristic algorithms proposed exclusively for (Kalpakis and Sherman, ; Ravada and Sherman, ; MacGregor Smith et al., ) consist in dividing the set of given points into subsets, then applying Smith's exact algorithm (Smith, ) and finally concatenate the exact solutions to find a heuristic solution. Heuristic algorithms for based on simulated annealing (Lundy, ), GRASP (Rocha, ), tabu search (Montenegro et al., ), and microcanonic algorithms (Montenegro et al., ), as well as an extended version of an ad hoc heuristic for (Chapeau‐Blondeau et al., ) to (Montenegro et al., ), have been proposed.…”

Self‐organized trail systems in groups of humans

“…MSTs feature new points, called Steiner Points, in addition to the original destination points. Soap films frequently create patterns that minimize surface tension, resulting in MST configurations [6]. Figure 3 show a device that can be easily constructed to study soap film formation.…”

Concatenation-Based Greedy Heuristics for the Euclidean Steiner Tree Problem

“…Finally, Chapeau-Blondeau et al [7] recently suggested a new O(n log n) heuristic. The (normalized) running times reported are slightly higher than those for SLL and SLL + and the average reduction for n = 1000 is only 2.78%.…”