A genetic approach with a simple fitness function for sorting unsigned permutations by reversals

Sorting unsigned permutations by reversals is an important and difficult problem in combinatorial processing of permutations with important applications in bio-informatics for the interpretation of the evolutionary relationship between organisms. Since it was shown that the problem is NP-hard many approximation and a few evolutionary algorithms were proposed. In this paper we propose a new genetic algorithm approach that uses modified crossover and mutation operators adapted to the problem. Instead previous genetic algorithmic approaches, the proposed algorithm uses a very simple fitness function that can be linearly computed in the size of the permutation and updated in constant time, for each individual in each generation. In order to compare the accuracy of the computed solutions, an 1.5 approximation ratio algorithm was developed by fixing Christie's approximation method. The results showed that on average the proposed genetic approach produces competitive results in relation with the ones given by the 1.5-approximation algorithm. Additionally, it has been observed that for permutations of all sizes, that were randomly generated, it was alway possible to compute better solutions with the genetic than with the approximate approach and that the difference obtained for these cases is greater than the ones obtained in the cases in which the genetic approach has a worse behavior than the approximate one.