Approximating the double-cut-and-join distance between unsigned genomes

Abstract

In this paper we study the problem of sorting unsigned genomes by double-cut-and-join operations, where genomes allow a mix of linear and circular chromosomes to be present. First, we formulate an equivalent optimization problem, called maximum cycle/path decomposition, which is aimed at finding a largest collection of edge-disjoint cycles/AA-paths/AB-paths in a breakpoint graph. Then, we show that the problem of finding a largest collection of edge-disjoint cycles/AA-paths/AB-paths of length no more than l can be reduced to the well-known degree-bounded k-set packing problem with k = 2l. Finally, a polynomial-time approximation algorithm for the problem of sorting unsigned genomes by double-cut-and-join operations is devised, which achieves the approximation ratio for any positive ε. For the restricted variation where each genome contains only one linear chromosome, the approximation ratio can be further improved to

Multiplicatively Exact Algorithms for Transformation and Reconstruction of Directed Path-Cycle Graphs with Repeated Edges

“…In ref. [10], a multiplicative error of at most 13/9 + ε was proved for this algorithm, where n is the size of the initial pair of graphs, and ε is any strictly positive number, the algorithm runtime being of the order of n O(ε −2.6 ) .…”

“…An undirected structure is a structure with all directions of edges eliminated-it can be defined directly, without using directed structures. In [10], for equal content of undirected structures and DCJ operations, the problem of finding the shortest transformation (in the number of used operations) of one such structure into another by an algorithm with good estimates of the accuracy and runtime was solved. In [10], a breakpoint graph A * B for two undirected structures A and B was defined.…”

“…In [10], for equal content of undirected structures and DCJ operations, the problem of finding the shortest transformation (in the number of used operations) of one such structure into another by an algorithm with good estimates of the accuracy and runtime was solved. In [10], a breakpoint graph A * B for two undirected structures A and B was defined. Specifically, its vertices are names of edges from A and B, and its edges connect names if they are assigned to adjacent edges in A or in B, and these edges are labeled, respectively, with the names A or B (see Figure 3).…”

“…In ref. [10], a ch-final graph (ch-final form) is defined. It consists of isolated vertices, vertices where an A-loop and a B-loop are joined, two different vertices connected by two pairs of A-and B-edges, and cycles AB of length 2 which are successively joined into a path or cycle of any length (see Figure 4).…”

Approximating the edit distance for genomes with duplicate genes under DCJ, insertion and deletion

“…We design an approximation algorithm by using techniques employed on the problem of B REAKPOINT G RAPH D ECOMPOSITION [5,6,18-20]. The basic idea is to find the maximum number of vertex-disjoint helpful cycles of length 4 in the adjacency graph.…”