On the Complexity of Rearrangement Problems under the Breakpoint Distance

TL;DR

This paper investigates the computational complexity of rearrangement problems under the breakpoint distance, revealing polynomial solutions for some problems and NP-hardness for others, including the small phylogeny problem for four species.

Contribution

It improves algorithms for the median problem and establishes NP-hardness of the small phylogeny problem and halving problem in breakpoint models, clarifying their computational difficulty.

Findings

Median problem is equivalent to maximum bipartite matching.

Small phylogeny problem is NP-hard for 4 species.

Halving problem is NP-hard in unichromosomal and multilinear models.

Abstract

We study complexity of rearrangement problems in the generalized breakpoint model and settle several open questions. The model was introduced by Tannier et al. (2009) who showed that the median problem is solvable in polynomial time in the multichromosomal circular and mixed breakpoint models. This is intriguing, since in most other rearrangement models (DCJ, reversal, unichromosomal or multilinear breakpoint models), the problem is NP-hard. The complexity of the small or even the large phylogeny problem under the breakpoint distance remained an open problem. We improve the algorithm for the median problem and show that it is equivalent to the problem of finding maximum cardinality non-bipartite 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 4 species (a quartet phylogeny). In other words, while finding an ancestor for 3 species is easy, finding two ancestors for 4 species is already hard. We also show that, in the unichromosomal and the multilinear breakpoint model, the halving problem is NP-hard, thus refuting the conjecture of Tannier et al. Interestingly, this is the first problem which is harder in the breakpoint model than in the DCJ or reversal models.