Sorting circular permutations by reversal

Unsigned circalar permutations are used to represent tours in the traveling salesman problem as well as the arrangement of gene loci in circular chromosomes. The minimum number of segment reversals required to transform one circular permutation into another gives some measure of distance between them which is useful when studying the 2-opt local search landscape for the traveling salesman problem, and, when determining the phylogeny of a group of related organisms. Computing this distance is equivalent to sorting by (a minimum number of) reversals. In this paper we show that sorting circular permutations by reversals can be reduced to the same problem for linear reversals, and that it is NP-hard. These results suggest that for most practical purposes any computational tools available for reversal sort of linear permutations will be sufficiently accurate. These results entail the development of the algebraic machinery for dealing rigorously with circular permutations.