An Approximation Algorithm for Sorting by Bounded Singleton Moves

Abstract

Sorting permutations by block moves is a fundamental combinatorial problem in genome rearrangements. The classic block move operation is called transposition, which switches two consecutive 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 block 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 [10], which requires the total length of blocks switched bounded by 3. By exploring some properties of this problem, we devise a $\frac{9}{5}$-approximation algorithm for $c=3$. Our algorithm can be extended to any constant $c\ge 3$, guaranteeing an approximation factor of $\frac{3c}{5}$.