Cycle decomposition

Abstract: We analyze the strong relationship among three combinatorial problems, namely, the problem of sorting a permutation by the minimum number of reversals (MIN-SBR), the problem of finding the maximum number of edge-disjoint alternating cycles in a breakpoint graph associated with a given permutation (MAX-ACD), and the problem of partitioning the edge set of an Eulerian graph into the maximum number of cycles (MAX-ECD). We first illustrate a nice characterization of breakpoint graphs, which leads to a linear-time algorithm for their recognition. This characterization is used to prove that MAX-ECD and MAX-ACD are equivalent, showing the latter to be NP-hard. We then describe a transformation from MAX-ACD to MIN-SBR, which is therefore shown to be NP-hard as well, answering an outstanding question which has been open for some years. Finally, we derive the worst-case performance of a well-known lower bound for MIN-SBR, obtained by solving MAX-ACD, discussing its implications on approximation algorithms for MIN-SBR.

Abstract: We consider two generalizations of signed Sorting Bg 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 Dee 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 nee 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 Zapproximation algorithms for two general problems which generalize MSBR and Tree SBR, respectively. To our knowledge, this work is the f?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.