Maximum Flow Without the Outer IPM
We show that the balancing weights technique of Li (2026) actually produces an approximate *pseudo-circulation* of a directed, capacitated graph in $m^{1+o(1)}$ time. Together with standard flow techniques, we obtain an $m^{1+o(1)}$ time maximum flow algorithm that avoids the interior-point method framework of recent almost-linear time algorithms (Chen et al. FOCS 2022, van den Brand et al. FOCS 2024).
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