The directed flow-cut gap is at most n^{1/3 + o(1)}.
Negative-weight single-source shortest paths in near-linear time: Now faster! In2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 515–538
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Improved Upper Bounds for the Directed Flow-Cut Gap
The directed flow-cut gap is at most n^{1/3 + o(1)}.