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 $m=\Theta(n^2)$ [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.

💡 Research Summary

The paper addresses the problem of identifying all “allowed edges” in a bipartite graph G = (V, E), i.e., edges that belong to at least one maximum matching. The authors show that, given any maximum matching M, all allowed edges can be enumerated in linear time O(|V| + |E|). Consequently, the overall complexity of the problem reduces to the time needed to compute a single maximum matching, which is O(√n · m) using the classic Hopcroft–Karp algorithm, or O((n/ log n)^{1/2} · m) for dense graphs (m = Θ(n²)) as per Alt et al. This improves upon the previous best deterministic bound O(n · m) (Costa 1994 for bipartite graphs, Carvalho & Cheriyan 2005 for general graphs) and also beats the randomized Õ(n^{2.376}) approaches of Rabin–Vazirani and Cheriyan when m = O(n^r) with r < 1.876.

The core technical contribution consists of two parts. First, for balanced bipartite graphs that admit a perfect matching, the authors prove that an edge is allowed iff it lies on an alternating cycle with respect to the given perfect matching, or it is itself part of the matching. By constructing a directed graph H whose vertices correspond to the left side of the bipartition and whose arcs (u_i, u_j) exist whenever (v_i, v’_j)∈E with i ≠ j, the problem reduces to detecting which arcs belong to a directed cycle. This is achieved by computing the strongly connected components (SCCs) of H (using Tarjan’s or Cheriyan‑Mehlhorn‑Gabo­w algorithms) and marking all edges whose endpoints lie in the same SCC. The perfect‑matching edges are trivially allowed. This procedure (Algorithm 2) runs in O(n + m).

Second, the authors extend the method to arbitrary bipartite graphs where the maximum matching size is t ≤ |V₁|. Vertices covered by M are called “upper” and the rest “lower”. They observe that any edge incident to a lower vertex is automatically allowed (otherwise M would not be maximal), and that there are no lower‑lower edges. Hence the problem reduces to the induced subgraph G_u on the upper vertices. Within G_u, allowed edges split into two categories: (I) those that belong to an alternating cycle in G_u (found again via SCCs), and (II) those that are not in any alternating cycle but can be placed in a maximum matching only together with a lower edge. The latter are precisely the edges that lie on a left‑ or right‑augmented alternating path (an alternating path that starts or ends at a lower vertex). Theorem 2.7 proves the equivalence. Detecting such paths can be done by a simple BFS/DFS from each lower vertex into G_u, marking any traversed upper edge as allowed of type II.

Algorithm 1 combines these steps: first compute a maximum matching M (using Hopcroft–Karp or any other method), then apply the linear‑time routine to label all allowed edges. The total running time is dominated by the matching phase, i.e., O(√n · m) in the worst case, while the subsequent labeling is O(n + m). The authors also discuss a dynamic setting where edges are inserted or deleted; by maintaining SCCs incrementally, updates can be handled in sublinear time per operation. Additional results in the appendix show that in regular bipartite graphs every edge is allowed, and they provide bounds on the size of the allowed‑edge set.

Overall, the paper delivers a conceptually simple, deterministic, and practically efficient algorithm for a fundamental problem in matching theory. Its linear‑time post‑processing makes it attractive for applications such as matchmaking services, privacy‑preserving data release, recommendation systems, and any domain where one needs to know which potential connections can appear in an optimal pairing. Future work may explore extensions to non‑bipartite graphs, tighter integration with faster matching algorithms, and large‑scale experimental evaluation.