The breakpoint distance for signed sequences

We consider the problem of estimating the rearrangement distance in terms of reversals, insertion and deletion between two genomes, G and H with possibly multiple genes from the same gene family. We define a notion of breakpoint distance for this problem, based on matching genes from the same family between G and H. We show that this distance is a good approximation of the edit distance, but NP-hard to compute, even when just one family of genes is non-trivial. We also propose a branch-and-cut exact algorithm for the computation of the breakpoint distance.