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 Björklund's last comment.

Can anyone help? Thank you in advance!

  • $\begingroup$ Schrijver's book on Combinatorial Optimization discusses several aspects of multiflows in Vol C. $\endgroup$ – Chandra Chekuri Jan 2 '16 at 4:55

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.

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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 Björklund's comment.

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