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Let's assume that for every $e\in E$ it holds that $c(e)$ is an integer. Does it mean that there exists a max flow $f$ that for every $e\in E$ it holds that $f(e)$ is an integer? It sounds obvious but I can't exactly say why it is true.

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    $\begingroup$ Have you ever heard of the max-flow min-cut theorem, the most famous and the most basic theorem in the theory of flow network? Have you ever heard of any algorithm that computes the maximum flow? $\endgroup$
    – John L.
    Jan 31, 2022 at 18:20

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The integral flow theorem says "yes".

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