Complexity of Cayley distance and other general metrics on permutation groups

Abstract

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.

Parallelization of genetic algorithms for sorting permutations by reversals over biological data

“…The results (average of number of reversals) of the second experiment is shown in the Table 2. Note that since we are looking for the minimum number of reversals the results provided by the MP GA are better than the previously obtained by the standard GA, though as previously mentioned, since the average of sorting by reversals is unknown [16], it is not possible to stablish the precision of these results.…”

“…The problem of sorting unsigned permutations by reversals (SUPR) addressed in this paper, was shown to be N P-hard in [11], but it is unknown whether it is N P-complete and also is unknown what is the average reversal distance of permutations of length n [16]. Approximation algorithms were developed in [6,8,12,15] of respective approximation radios 2.0, 1.75, 1.5 and 1.375, being the last algorithm of purely theoretical interest only.…”