# Does multi-commodity flow problem (MCF) block cycling flows?

I am working with multi-commodity flow at integral flow with unit demand for each operation.

Since I use the MCF formulation to model path finding, in my scenario cycles are just a waste of resources. I often read that MCF models path finding problems, without making particular assumptions on the graph (which in my case is undirected).

However, I can't see why an MCF solver shouldn't send a flow within a cycle to maximize the objective function, even in my scenario with unit demand.

• In other words, assume $\epsilon \in [0,1]$ being the lower-bound. The LP solver gives a solution which can't be worse than $f^{OPT}*\epsilon$? Also, if $f^{OPT}$ doesn't exist, the LP solver will give empty solution. Oct 13 at 16:43