The directed flow-cut gap is at most n^{1/3 + o(1)}.
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms.Journal of the ACM (JACM), 46(6):787–832
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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)}.