Approximation Algorithms for Sorting by Bounded Singleton Moves

Sorting permutations by block moves is a fundamental combinatorial problem in genome rearrangements. The classic block move operation is called transposition, which switches two adjacent blocks, or equivalently, moves a block to some other position. But large blocks movement rarely occurs during real evolutionary events. A natural restriction of transposition is to bound the length of the blocks to be switched. In this paper, we investigate the problem called sorting by bounded singleton moves, where one of the switched blocks is exactly a singleton while the other is of length at most c. This problem generalizes the sorting by short block moves problem proposed by Heath and Vergara [11], which requires the total length of blocks switched bounded by 3. When c=3, we devise a 95-approximation algorithm for an arbitrary permutation, and a 127-approximation algorithm for a woven double-strip permutation. Our algorithms can be slightly extended to solve the sorting by c-bounded singleton moves problem for any constant c≥3, guaranteeing an approximation factor of 3c5 and 4c7 for arbitrary permutations and woven double-strip permutations respectively, just by exploiting a new lower bound of sorting by c-bounded singleton moves.