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.
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.
