Formulations and Hardness ofMultiple Sorting

We consider two generalizations of signed Sorting By Reversals (SBR), both aimed at formalizing the problem of reconstructing the evolutionary history of a set of species. In particular, we address Multiple SBR, calling for a signed permutation at minimum reversal distance from a given set of signed permutations, and Tree SBR, calling for a tree with the minimum number of edges spanning a given set of nodes in the complete graph where each node corresponds to a signed permutation and there is an edge between each pair of signed permutations one reversal away from each other. We describe a graph-theoretic relaxation of MSBR, which is the counterpart of the so-called alternating-cycle decomposition relaxation for SBR, illustrating a convenient mathematical formulation for this relaxation. Moreover, we use this relaxation to show that, even if the number of given permutations equals 3, MSBR is NP-hard, and hence so is Tree SBR. In fact, we show that the two problems are APX-hard, i.e. they do not have a polynomial-time approximation scheme unless P=NP. Finally, we mention known 2-approximation algorithms for two general problems which generalize MSBR and Tree SBR, respectively. To our knowledge, this work is the rst one discussing the complexity of MBSR (and Tree SBR), as well as potential solution approaches to the problem based on the use of a tight relaxation.