Finding all minimum-cost perfect matchings in Bipartite graphs

The Hungarian method is an e cient algorithm for nding a minimal cost perfect matching in a weighted bipartite graph. This paper describes an e cient algorithm for nding all minimal cost perfect matchings. The computational time required to generate each additional perfect matching is O(n(n + m)); and it requires O(n+m) memory storage. This problem can be solved by algorithms for nding the Kth-best solution of assignment problems. However, the memory storage required by the known algorithms grows in proportion to K; and hence it may grow exponentially in n: So, our specialized algorithm has a considerable advantage in memory requirement over the previous more general algorithms for Kth-best assignment problems. Here we will show that the enumeration of all minimal cost perfect matchings can be reduced to the enumeration of all perfect matchings in some bipartite graph. Therefore our algorithm can be seen as an algorithm for enumerating all perfect matchings in a given bipartite graph.