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Finding All Allowed Edges in a Bipartite Graph

We consider the problem of finding all allowed edges in a bipartite graph $G=(V,E)$, i.e., all edges that are included in some maximum matching. We show that given any maximum matching in the graph, it is possible to perform this computation in linear time $O(n+m)$ (where $n=|V|$ and $m=|E|$). Hence, the time complexity of finding all allowed edges reduces to that of finding a single maximum matching, which is $O(n^{1/2}m)$ [Hopcroft and Karp 1973], or $O((n/\log n)^{1/2}m)$ for dense graphs with $m=Θ(n^2)$ [Alt et al. 1991]. This time complexity improves upon that of the best known algorithms for the problem, which is $O(nm)$ ([Costa 1994] for bipartite graphs, and [Carvalho and Cheriyan 2005] for general graphs). Other algorithms for solving that problem are randomized algorithms due to [Rabin and Vazirani 1989] and [Cheriyan 1997], the runtime of which is $\tilde{O}(n^{2.376})$. Our algorithm, apart from being deterministic, improves upon that time complexity for bipartite graphs when $m=O(n^r)$ and $r<1.876$. In addition, our algorithm is elementary, conceptually simple, and easy to implement.