I'm searching for an example digraph for the multi-commodity minimum cost flow problem with integer demand. There shouldn't be an integer but fractional optimal solution. I found here a similar question but it considers an undirected graph or to be precise I don't catch Andreas Bjrklund's last comment.
Can anyone help? Thank you in advance!
You can use the example for an undirected edge with unit capacities, but replace each undirected edge from A to B with a set of directed edges that look like this.
Each edge has a unit capacity. And if you have $x$ units of flow from A to B and $y$ units of flow from B to A, you can route it as long as $x+y \leq 1$.
This is the gadget referred to in Andreas Bjrklund's comment.
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