I am working on a flow problem with side constraints. More specifically, I have a usual flow problem, with constraints that require some arcs to have exactly one unit of flow on them. This makes the problem "hard", as the classic formulation for flows may leed to subtours.
The structure of my graph is such that the subtours can only have an even number of arcs larger than 4. So if I impose that I have exactly 5 arcs, with one leaving the source, and one entering the sink, I eliminate solutions with subtours.
I am interested in solutions with 7 arcs, and am trying to come up with a "simple" constraint (no MTZ constraints, nor typical subtour elimination constraints from the TSP, nor no-good cuts) that eliminates solutions with a subtour of 4 arcs, and a path from source to sink with 2 intermediate nodes.
Any ideas ?
For more clarity, here is an example of a solution that I would like to eliminate (the blue triangle indicates that I need a unit of flow on this arc):
And here is a valid solution with 7 arcs: