A 2-Approximation Scheme for Sorting Signed Permutations by Reversals, Transpositions, Transreversals, and Block-Interchanges

We consider the problem of sorting signed permutations by reversals, transpositions, transreversals, and block-interchanges and give a 2-approximation scheme, called the GSB (Genome Sorting by Bridges) scheme. Our result extends 2-approximation algorithm of He and Chen [12] that allowed only reversals and block-interchanges, and also the 1.5 approximation algorithm of Hartman and Sharan [11] that allowed only transreversals and transpositions. We prove this result by introducing three bridge structures in the breakpoint graph, namely, the L-bridge, T-bridge, and X-bridge and show that they model "proper" reversals, transpositions, transreversals, and block-interchanges, respectively. We show that we can always find at least one of these three bridges in any breakpoint graph, thus giving an upper bound on the number of operations needed. We prove a lower bound on the distance and use it to show that GSB has a 2-approximation ratio. An ${\text{O(n}}<^>{3})$O(n3) algorithm called GSB-I that is based on the GSB approximation scheme presented in this paper has recently been published by Yu, Hao, and Leong in [17] . We note that our 2-approximation scheme admits many possible implementations by varying the order we search for proper rearrangement operations.