Date: 29 May 2005

On steiner trees and genetic algorithms

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In this paper the application of a Genetic Algorithm (GA) to the Steiner tree problem is described. The performance of the GA is compared to that of the Simulated Annealing Algorithm (SA) and one of the best conventional algorithms given by Rayward-Smith and Clare [1] (RCA).

Particular attention has been paid to find an optimal setting of the parameters and operators of the GA. A mutation probability P M=0.01 and a crossover probability P C=0.5 have been obtained according to the values found by Grefenstette [2]. An optimal population size has been chosen using a heuristic comparable to that of Goldberg

In addition the application of two problem specific heuristics is discussed, the removal of Steiner points of degree <3 and a local relaxation of the tree. A speedup of ≈20 was obtained if these heuristics were applied to the computation of the fitness only without changing the individuals themselves. For a fair comparison, these heuristics have been applied to the GA, the SA, and, in contrast to its original definition, to the RCA.

According to our results, all three algorithms find the optimum and converge equally fast, i.e., GA and SA need not more function evaluations than the RCA, that has even been improved by the relaxation heuristic. The GA reaches the optimum in a small number of generations which is considered the reason why it did not show an even better performance.