Finding all minimum‐cost perfect matchings in Bipartite graphs

Abstract

AbstractThe Hungarian method is an efficient algorithm for finding a minimal‐cost perfect matching in a weighted bipartite graph. This paper describes an efficient algorithm for finding 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 finding 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 precious 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.