Sorting by Multi-cut Rearrangements

Abstract

Let S be a string built on some alphabet $\varSigma$. A multi-cut rearrangement of S is a string $S'$ obtained from S by an operation called k-cut rearrangement, that consists in (1) cutting S at a given number k of places in S, making S the concatenated string $X_1\cdot X_2\cdot X_3\ldots X_k\cdot X_{k+1}$, where $X_1$ and $X_{k+1}$ are possibly empty, and (2) rearranging the $X_i$s so as to obtain $S'=X_{\pi (1)}\cdot X_{\pi (2)}\cdot X_{\pi (3)}\ldots X_{\pi (k+1)}$, $\pi$ being a permutation on $1,2\ldots k+1$ satisfying $\pi (1)=1$ and $\pi (k+1)=k+1$. Given two strings S and T built on the same multiset of characters from $\varSigma$, the Sorting by Multi-cut Rearrangements (SMCR) problem asks whether a given number $\ell$ of $k$-cut rearrangements suffices to transform S into T. The SMCR problem generalizes and thus encompasses 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 in massive simultaneous rearrangements. In this paper, we study the SMCR problem from an algorithmic complexity viewpoint, and provide, depending on the respective values of k and $\ell$, polynomial-time algorithms as well as NP-hardness, FPT-algorithms, W[1]-hardness and approximation results, either in the general case or when S and T are permutations.

GF was partially supported by the PHC Procope program 17746PC

GJ was partially supported by the PHC Procope program 17746PC

CK was partially supported by the DAAD Procope program 57317050.