The $z$-matching problem on bipartite graphs

Abstract

The $z$-matching problem on bipartite graphs is studied with a local algorithm. A $z$-matching ($z \ge 1$) on a bipartite graph is a set of matched edges, in which each vertex of one type is adjacent to at most $1$ matched edge and each vertex of the other type is adjacent to at most $z$ matched edges. The $z$-matching problem on a given bipartite graph concerns finding $z$-matchings with the maximum size. Our approach to this combinatorial optimization are of two folds. From an algorithmic perspective, we adopt a local algorithm as a linear approximate solver to find $z$-matchings on general bipartite graphs, whose basic component is a generalized version of the greedy leaf removal procedure in graph theory. From an analytical perspective, in the case of random bipartite graphs with the same size of two types of vertices, we develop a mean-field theory for the percolation phenomenon underlying the local algorithm, leading to a theoretical estimation of $z$-matching sizes on coreless graphs. We hope that our results can shed light on further study on algorithms and computational complexity of the optimization problem.Comment: 15 pages, 3 figure