A balancing-weights technique lets a randomized augmenting-paths algorithm compute max flow in directed graphs in m + nF time, matching the undirected case.
On finding maximum flows in networks with special structure and some applications
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Balancing Weights, Directed Sparsification, and Augmenting Paths
A balancing-weights technique lets a randomized augmenting-paths algorithm compute max flow in directed graphs in m + nF time, matching the undirected case.