# Linear programming and network flow

I would like some hint in this homework question. I have to write the max-flow problem (with souce $$s$$ and sink $$t$$) as a linear program. I have to do this by defining variables on each $$s - t$$ path, which is different than the typical approach of considering as variables the flow of each edge.

So far, I thought on the following. Let $$f_i$$ be the flow of each $$s-t$$ path, then the objective would be $$\max_{f_i} \sum_i f_i$$, subject to every $$f_i$$ is less or equal that the capacity of each edge on that path, plus the flow convservation constraints on each vertex. Is this correct?