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 O(n 3 ) 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.