A (1+ε)-approximation algorithm for sorting by short block-moves

Sorting permutations by operations such as reversals and block-moves has received much attention because of its applications in the study of genome rearrangements. A short block-move is an operation on a permutation that moves an element at most two positions away from its original position. In this paper, we investigate the problem of finding a minimum-length sorting sequence of short block-moves for a given permutation, and devise a (1+ε)-approximation algorithm for this problem, where ε is the number of elements divided by the number of inversions in the permutation. The algorithm mostly relies on a new structure in the permutation graph called an umbrella, which can be optimally sorted in O(n2) time.