Toward minimal restriction of genetic encoding and crossovers for the two-dimensional Euclidean TSP

In the field of evolutionary algorithms (EAs), many operators have been introduced for the traveling salesman problem. Most encoding schemes have various restrictions that often result in a loss of information contained in problem instances. We suggest a new chromosomal encoding scheme that pursues minimal information loss and a crossover scheme with minimal restriction for the two-dimensional (2-D) Euclidean TSP. The most notable feature of the suggested crossover is that it uses the 2-D tour images themselves for chromosomal cutting. We prove the theoretical validity of the new crossover by an equivalence-class analysis. The proposed encoding/crossover pair outperformed both distance-preserving crossover and edge-assembly crossover, which represent two state-of-the-art crossovers in the literature. We also tested its performance on a mixed framework that incorporates a large-step Markov chain technique into the framework of a traditional EA.