Sorting by Multi-Cut Rearrangements

A multi-cut rearrangement of a string S is S? obtained from by an operation called k-cut rearrangement, that consists (1) cutting at given number k places in S, making the concatenated X1·X2·X3·…·Xk·Xk+1, where X1 and Xk+1 are possibly empty, (2) rearranging Xis so as to obtain S?=X?(1)·X?(2)·X?(3)·…·X?(k+1), ? being permutation on 1,2,…,k+1 satisfying ?(1)=1 ?(k+1)=k+1. Given two strings T built same multiset characters alphabet ?, Sorting Multi-Cut Rearrangements (SMCR) problem asks whether ? rearrangements suffices transform into T. The SMCR generalizes several classical genomic problems, such Transpositions Block Interchanges. It may also model chromoanagenesis, recently discovered phenomenon consisting massive simultaneous rearrangements. In this paper, we study algorithmic complexity viewpoint. More precisely, investigate its parameterized complexity, well approximability, general case or when permutations.