Sorting Permutations by λ-Operations

For estimating the evolutionary distance between genomes of two different organisms, many sorting permutation problems have emerged. A well accepted way to do this is considering the smallest sequence of rearrangement events { mutations which affect large portions of the genomes { that transform one genome into the other. In these problems, both genomes are represented as permutations of integer numbers, but one of them can be represented as the identity permutation, so that the problem is reduced to sort a permutation. Moreover, rearrangement models define which type of operations (rearrangement events) can be applied over a permutation in order to modify it. Reversals, which are operations that revert a genome segment, and transpositions, which are operations that swap two adjacent genome segments, are two of the most studied types of rearrangements in the literature. In this paper, we consider rearrangement models that allow reversals, transpositions, and both operations together. Since there exist evidences that large mutations rarely occur, we add a restriction with biological relevance: any operation can affect at most λ elements of a permutation. For this variation of sorting permutation problem, we present approximation algorithms with approximation factors based on the size of the permutation and/or on λ, for both signed and unsigned permutations (which represent known and unknown gene orientations, respectively)