In this talk we study the problems of sorting signed permutations
by reversals (SBR) and sorting unsigned permutations by
transpositions (SBT), which are fundamental problems in
computational molecular biology. While a polynomial-time solution
for SBR is known, SBT has been open for more than a decade and is
considered a major open problem. We outline a novel combinatorial
framework for exploring SBT, and introduce a graphical model which
is analogous to the interleaving graph of the Hannenhalli-Pevzner
Theory for SBR. We further generalize this model, and connect it
to an algebraic question about matrices over rings. We discover
that this general model is in fact a common framework for both SBR
and SBT (and other rearrangement problems). This model gives deep
insight on these problems and sheds new light on old results. We
are thus left with a problem of combinatorial-algebraic nature
which is fundamental to several genome rearrangement problems. We
give a variety of results about this problem and pose many open
problems.
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