Deterministic and Probabilistic Algorithms for Maximum Bipartite Matching Via Fast Matrix Multiplication

Let G = (S, T, E) be a bipartite-graph, where S U T is the set of nodes (S n T = 8) and E is the set of edges, E c S X T. Let S = {ur , . . . . II,}, T = {VI, . . . . vt} (A t), and 1 E I= e. An (S, T) matching is a subset M of E such that no two edges in M have a common endpoint. A maximum matching is a matching of maximum cardinality. The set of nodes which take part in such a maximum matching is denoted by Nodes(G) and the cardinality of the matching is denoted by Card(G). Note that Nodes(G) is not unique, but Card(G) is unique. There is an O(e& algorithm to find a maximum matching [3 1. When e = O(st) (i.e., the graph is dense), O(e&) = Q(sl*s t). Let A = (aij) be an s X t matrix over a given ring. A matching of cardinality r in A is a set of r nonzero entries of A, with at most one entry chosen from each row and column. The cardinality of a maximal matching in A is denoted by Card(A). Let F be any field, and let G = (S, T, E) be given. With each edge (ui, vj) in E we associate a variable xij, and with the graph G we associate a matrix AG = (aii) defined by: