Complexity of Cayley distance and other general metrics on permutation groups

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, NP-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, an error is pointed out in one of the polynomial reductions applied in Pinch´s approach attempting to proof that the subgroup distance problem for Cayley distance is NP-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.