On the complexity of rearrangement problems under the breakpoint distance.

We study the complexity of rearrangement problems in the generalized breakpoint model of Tannier et al. and settle several open questions. We improve the algorithm for the median problem and show that it is equivalent to the problem of finding maximum cardinality nonbipartite matching (under linear reduction). On the other hand, we prove that the more general small phylogeny problem is NP-hard. Surprisingly, we show that it is already NP-hard (or even APX-hard) for a quartet phylogeny. We also show that in the unichromosomal and the multilinear breakpoint model the halving problem is NP-hard, refuting the conjecture of Tannier et al. Interestingly, this is the first problem that is harder in the breakpoint model than in the double cut and join or reversal models.