Sorting by Multi-Cut Rearrangements

A multi-cut rearrangement of a string S is a string S ' obtained from S by an operation called k-cut rearrangement, that consists of (1) cutting S at a given number k of places in S, making S the concatenated string X-1 center dot X-2 center dot X-3 center dot ... center dot X-k center dot Xk+1, where X-1 and Xk+1 are possibly empty, and (2) rearranging the Xis so as to obtain S '=X-pi(1)center dot X-pi(2)center dot X-pi(3)center dot ... center dot X pi(k+1), pi being a permutation on 1,2, horizontal ellipsis ,k+1 satisfying pi(1)=1 and pi(k+1)=k+1. Given two strings S and T built on the same multiset of characters from an alphabet sigma, the Sorting by Multi-Cut Rearrangements (SMCR) problem asks whether a given number l of k-cut rearrangements suffices to transform S into T. The SMCR problem generalizes several classical genomic rearrangements problems, such as Sorting by Transpositions and Sorting by Block Interchanges. It may also model chromoanagenesis, a recently discovered phenomenon consisting of massive simultaneous rearrangements. In this paper, we study the SMCR problem from an algorithmic complexity viewpoint. More precisely, we investigate its classical and parameterized complexity, as well as its approximability, in the general case or when S and T are permutations.