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Abstract-Translocation is a useful operation on strings with challenging questions in combinatorics of permutations and interesting applications in analysis of sequences. A translocation operation essentially is the interchange of prefixes and suffixes among two substrings of a string. For the case of genomes represented as strings, symbols represent genes and chromosomes are modeled as substrings of the genomes; thus, translocation is an operation that models the interaction between chromosomes among a genome. The translocation distance between two genomes is defined as the minimum number of translocations to convert one genome into the other and had been proved to be a meaningful manner of modeling the evolutive distance between organisms. The particular case of unsigned genomes, those in which the orientation of the genes are not considered, is particularly difficult, while the signed case, in which the orientation of genes is considered, has been proved to be polynomially decidable. This paper compiles a proof of the N P-hardness and presents an innovative GA approach to solve the unsigned translocation distance problem. As distinguished feature, the proposed GA uses as fitness function the translocation distance for randomly generated signed versions of the unsigned genomes. Experiments over randomly generated strings (synthetic chromosomes) show that the proposed GA approach compute answers that are better than those computed by an 1.5+ε-approximation algorithm, the latter also implemented as part of this work.

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Abstract-Permutation groups arise as important structures in group theory because many algebraic properties about them are well-known, which makes modeling natural phenomena by permutations of practical interest. Usability of the involved algebraic notions is illustrated by problems such as genome rearrangement by reversals for which it is well-known that for the case of unsigned and signed sorting by reversals the time complexity is, respectively, N P-hard and P. Reversal distance is a particular metric and in this work more general metrics on permutation groups are considered emphasizing on the Cayley distance. In particular, we point out an error in one of the polynomial reductions applied in Pinch's approach attempting to proof that the subgroup distance problem for Cayley distance is N P-complete and following his approach we present a simplified and correct proof of this fact. Although, recently a shorter and more general proof than Pinch's one was given by Buchheim, Cameron and Wu, we believe the correction of Pinch's proof presented in this paper is of great interest because it correctly relates the Cayley distance problem with a maximal routing problem giving an additional perspective in relation to Buchheim et al. recent proof from which only the usual logical satisfiability perspective of distance problems is observable.