Strongly Polynomial Parallel Maximum Flow Revisited

We study the maximum flow problem in directed networks with real capacities in the parallel setting. For a network with $n$ vertices and $m$ arcs, we show that a randomized parallel implementation of a variant of the strongly polynomial max-flow algorithm of Dadush, Orlin, Sidford, and Végh [SODA 2026] runs in $\tilde{O}(mn)$ work and $\tilde{O}(m)$ depth. This improves upon the previously described tradeoffs between work and depth for strongly polynomial parallel maximum flow algorithms: earlier $\tilde{O}(n^3)$-work algorithms have $\tilde{O}(n^2)$ depth [Shiloach and Vishkin, J. Algorithms 1982; Goldberg and Tarjan, J. ACM 1988], while the known $\tilde{O}(m)$-depth approach uses $\tilde{O}(m{n}^3)$ work [Orlin, Oper. Res. 1993].