# Network Flow - Minimum flow in a network

I have a directed graph G=(V,E) with a source s$$\in V$$ and a sink t$$\in V$$. There is a minimum capacity (lower bound) l $$_{e}$$ for each edge in G. There are no upper bounds on the edges.

In a course that I took, the professor told that to find a minimum flow -

1) We need to assign a large capacity to all edges and find flow f
2) Construct G $$_{1}$$ where all edges are reversed and each edge has capacity f$$_{e}$$ - l$$_{e}$$
3) We need to then find the max flow from t to s in G$$_{1}$$ that is f$$_{1}$$
4) Then, the minimum flow in G is f-f$$_{1}$$

My question is- Why can't we find a s to t path in G with the least value of l$$_{e}$$. The least value of l$$_{e}$$ would be the minimum flow that could be pushed through the network?

The $$(s,t)$$-path with the lowest flow demand needs $$1$$ unit of flow, but the minimum flow is $$4$$. And the $$(s,t)$$-path with minimum sums of capacities has a sum of $$2$$, so this also doesn't work.