Given a flow network $G=(V,E)$, source $s$ , sink $t$ and capacity function $c:E \to \mathbb{R}^+ \cup \{0\}$ ; as well an edge $e=(u,v) \in E$. I need to find an efficient algorithm which finds among all possible flows between $s$ and $t$ , a flow $f$ where $f(e)$ is the maximum flow possible on the edge $e$ for it.
I want to use Ford Fulkerson algorithm but instead of using BFSs one after the other and increase the flow, first use all the paths through $e$, and then after we don't find any, go on with any path available to $t$ from $s$, or something in this direction..
Edit: You can assume that the capacities are integers.
Thanks a lot!