Matrix tightness: A linear-algebraic framework for sorting by transpositions

Abstract

We study the problems of sorting signed permutations by reversals (SBR) and sorting unsigned permutations by transpositions (SBT), which are central problems in computational molecular biology. While a polynomial-time solution for SBR is known, the computational complexity of SET has been open for more than a decade and is considered a major open problem. In the first efficient solution of SBR, Hannenhalli and Pevzner [HP99] used a graph-theoretic model for representing permutations, called the interleaving graph. This model was crucial to their solution. Here, we define a new model for SET, which is analogous to the interleaving graph. Our model has some desirable properties that were lacking in earlier models for SET. These properties make it extremely useful for studying SET. Using this model, we give a linear-algebraic framework in which SET can be studied. Specifically, for matrices over any algebraic ring, we define a class of matrices called tight matrices. We show that an efficient algorithm which recognizes tight matrices over a certain ring, double struck M sign, implies an efficient algorithm that solves SBT on an important class of permutations, called simple permutations. Such an algorithm is likely to lead to an efficient algorithm for SBT that works on all permutations. The problem of recognizing tight matrices is also a generalization of SBR and of a large class of other "sorting by rearrangements" problems, and seems interesting in its own right as. We give an efficient algorithm for recognizing tight symmetric matrices over any fiejd of characteristic 2. We leave as an open problem to find an efficient algorithm for recognizing tight matrices over the ring M. © Springer-Verlag Berlin Heidelberg 2006.