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

Konstantin Gorbunov,Vassily Lyubetsky

doi:10.3390/math9202576

Abstract

For any weighted directed path-cycle graphs, a and b (referred to as structures), and any equal costs of operations (intermergings and duplication), we obtain an algorithm which, by successively applying these operations to a, outputs b if the first structure contains no paralogs (i.e., edges with a repeated name) and the second has no more than two paralogs for each edge. In finding the shortest sequence of operations to be applied to pass from a to b, the algorithm has a multiplicative error of at most 13/9 + ε, where ε is any strictly positive number, and its runtime is of the order of nO(ε−2.6), where n is the size of the input pair of graphs. In the case of no paralogs, equal sets of names in the structures, and equal operation costs, we have considered the following conditions on the transformation of a into b: all structures in them are from one cycle; all structures are from one path; all structures are from paths. For each of the conditions, we have obtained an exact (i.e., zero-error) quadratic time algorithm for finding the shortest transformation of a into b. For another list of operations (join and cut of a vertex, and deletion and insertion of an edge) over structures and for arbitrary costs of these operations, we have obtained an algorithm for the extension of structures specified at the leaves of a tree onto its interior vertices. The algorithm is exact if the tree is a star—in this case, structures in the leaves may even have unequal sets of names or paralogs. The runtime of the algorithm is of the order of nΧ + n2log(n), where n is the number of names in the leaves, and Χ is an easily computable characteristic of the structures in the leaves. In the general case, a cubic time algorithm finds a locally minimal solution.

Highlights

Directed StructuresAny discrete optimization algorithm is said to be exact if its output exactly coincides with one of the minima of that functional
By assigning each pair p with the number pno − pyes, we reduce the original problem, by a quadratic time algorithm, to the problem on the maximal weighted matching in a complete graph
The algorithm from Theorem 1 has a multiplicative error of at most 13/9 + ε, where ε is any strictly positive number, and its runtime is of the order of nO(ε−2.6), where n is the size of the input pair of structures

Summary

Introduction

Any discrete optimization algorithm (minimizing a function or a functional) is said to be exact if its output exactly coincides with one of the minima of that functional. Recall the definition of a breakpoint (sometimes referred to as common) graph a + b for structures a and b with equal content This is an undirected graph consisting of paths and cycles (not loops) whose edges are labeled with symbols a or b. Sets of names in a and b are the same (i.e., the structures have quasi-equal content), and only Cut and OM out of the four DCJ operations are allowed, together with an additional operation D of edge duplication, i.e., insertion of a new edge next to a given one, with the same name and direction (linear duplication), or adding an edge with the same name as a loop, a separate component of the structure (cyclic duplication). The problems of improving the below given upper bound on the accuracy of the algorithm (i.e., proximity of the algorithm output to the exact minimum), as well as of passing to three or more allowed paralogs of an edge in b, are open and apparently hard

Common Graph and Operations over It

Transformation of Structures with Duplications

Reconstructions of Structures along a Tree by SCJ Operations

Discussion