Genetic Operators, the Tness Landscape and the Traveling Salesman Problem
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Edge Recombination and Maximal Preservative Crossover (MPX) are two operators developed to preserve edge information for the Traveling Salesman Problem. Manderick et al. introduced the notion of a tness landscape to measure the tness correlation between parents and ospring under dierent recombination operators. The work on the tness landscape is extended by studying the interrelationship between the tness landscape, operator failure rates (in terms of non-inherited edges) and the eect of operator failure on tour length. The use of local improvement operators is also examined. The application of genetic based search to Traveling Salesman Problems (TSP) of several hundred cities has produced encouraging results. M uhlenbein [5], Ulder et al. [7], Gorges-Schleuter [2], and Eshelman [1] all report near optimal results on the Padberg 532-city problem. All of these approaches combine local search in the form of 2-Opt with genetic search. We look at two recombination operators, Edge and Maximal Preservative Crossover (MPX), developed especially for the TSP. To better understand the computational behavior of these genetic operators we examine their recombination behavior using various metrics, including the op metric introduced by Manderick et al. [4] to look at the tness landscape. 1.1. Alphabet cardinality One thing that distinguishes the application of genetic algorithms to permutation problems such as the TSP from other optimization problems is the nature of the encoding. A great deal of eort has been expended creating crossover operators that recombine sequence permutations while maintaining feasibility and transferring as much critical ad-jacency information from parents to ospring as possible. However, alphabet cardinality is also a critical issue, especially as the size of the permutation problem becomes larger. Each tour in a TSP is a Hamiltonian cycle on a fully connected graph where each city is a vertex. The fully connected graph for an N city TSP has (N 2 0 N)=2 edges. For a 100 city problem, the corresponding fully connected graph has 4950 edges. If each tour sampled 100 unique edges, then at least d(N 01)=2e, or 50 tours would be needed to cover the graph. By covering the graph, we mean that every edge in the graph is included at least once in the population of N/2 tours.