We present an algorithm that enumerates all the perfect matchings in a given bipartite graph G = (V,E). Our algorithm requires a constant amortized time to visit one perfect matching of G, in contrast to the current fastest algorithm, published 25 years ago by Uno, which requires O(log |V|) time. To facilitate the listing of all edges in a visited perfect matching, we develop a variant of arithmetic circuits, which may have broader applications in future enumeration algorithms. Consequently, a visited perfect matching is represented within a binary tree. Although it is more common to provide visited objects in an array, we present a class of graphs for which achieving constant amortized time is not feasible in this case.
Jiří Fink (2025) studied this question.