Minimize bottleneck in flow network

Question

Let $G=(V,E)$ be a flow network with two vertices $s,t$ also each edge has its capacity equal to $\infty$. Our goal is to transfer a flow of size $C$ from $s$ to $t$ so that minimize an edge that has highest flow value in it.

I guess by $O(1) $ times using the Ford-Fulkerson we can minimize such an edge. First let $k$ be the degree of $s$, then I set the capacity of each edges that outs from $s$ to $\frac{c}{k}$. Now at this step I get stuck. Any hint about how I can do it will be helpful.

Answer 1

Let us reverse the question. In other words, consider the dual question.

Instead of minimizing the highest flow value of all edges that can transfer a flow of size $C$, let us compute the maximum flow value of the network if every edge has capacity $x$.

Suppose that maximum flow value is $m$, which can be computed by the Ford–Fulkerson algorithm. Then the highest flow value of all edges that can transfer a flow of size $C$ must be at least $x\cdot\frac Cm$, assuming $m>0$. The case $m=0$ is easy.