A hybrid genetic algorithm (GA) which combines the global search power of GA with the local search power of a local optimization algorithm is described for the graph coloring problem (GCP). Each solution of the GCP, which is called phenotype, is represented by a set of isomorphic genotypes conceptually. Then, a metric function between two phenotypes is defined by the least Hamming distance between the corresponding sets of isomorphic genotypes. The phenotypic distance is useful to analyze and control the behavior of genotypes in the search space from the view point of the problem space. A new crossover technique named harmonic crossover is proposed for the GCP. The phenotypic distance between two parents is considered in the harmonic crossover for preserving their common characteristics. Furthermore, the phenotypic distance between two parents is also used to predict promising regions in the problem space. In the proposed hybrid GA for the GCP, the local optimization algorithm is applied only in the most promising regions restrictedly and intensively. Consequently, the run of the local optimization algorithm does not hinder the performance of GA in its progress of global search.
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