An Efficient Implicit OBDD-Based Algorithm for Maximal Matchings

The maximal matching problem, i.e., the computation of a matching that is not a proper subset of another matching, is a fundamental optimization problem and algorithms for maximal matchings have been used as submodules for problems like maximal node-disjoint paths or maximum flow. Since in some applications graphs become larger and larger, a research branch has emerged which is concerned with the design and analysis of implicit algorithms for classical graph problems. Inputs are given as characteristic Boolean functions of the edge sets and problems have to be solved by functional operations efficiently supported by the chosen data structure. An OBDD is a well-known data structure for Boolean functions and sometimes capable to take advantage over the presence of regular substructures which may lead to sublinear graph representations. As a result OBDD-based algorithms are used as a heuristic approach to handle very large graphs. Here, an implicit OBDD-based maximal matching algorithm is presented that uses only a polylogarithmic number of functional operations with respect to the number of vertices of the input graph. In order to investigate the algorithm’s behavior on large and structured networks, it has been analyzed on grid graphs and it has been shown that the overall running time and the space requirement is also polylogarithmic.