Theoretical Computer Science
graph-theory examples multicommodity-flow
Updated Wed, 25 May 2022 02:17:10 GMT

# Fractional but not integer multi-commodity minimum cost flow

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!

## Solution

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.