Signed genome rearrangement by reversals and transpositions: Models and approximations?

An important problem in computational molecular biology is the genome rearrangement using reversals and transpositions. Analysis of genome evolving by inversions and transpositions leads to a combina- torial optimization problem of sorting by reversals and transpositions, i.e., sorting of a permutation using reversals and transpositions of arbi- trary fragments. The reversal operation works on a single segment of the genome by reversing the selected segment. Two kinds of transpositions have been studied in the literature. The first kind of transposition oper- ation deletes a segment of the genome and insert it into another position in the genome. The second kind of transposition operation deletes a seg- ment of the genome and insert its inverse into another position in the genome. Both transposition operations can be viewed as operations work- ing on two consecutive segments. A third transposition operation working on two consecutive segments is introduced which, together with reversal and the first two kinds of transposition operations, forms the complete set of operations on two consecutive segments. In this paper, we study the sorting of a signed permutation by reversals and transpositions. By allowing only the first kind of transpositions, or the first two kinds of transpositions, or all three kinds of transpositions, we have three prob- lem models. After establishing a common lower bound on the numbers of operations needed, we present a unified 2-approximation algorithm for all these problems. Finally, we present a better 1:75-approximation for the third problem.