Consider some weighted, directed graph $G=(V,E)$. How can we find the edge that is the maximum of all minimum values along paths from $s$ to $t$?
For example, if there are two paths from $s$ to $t$ and the first path has minimum edge $2$ and the second has minimum edge $5$, we return the path that has minimum edge $5$ since it is the path with a minimum-capacity edge maximized. Similarly, if there are three paths from $s$ to $t$ with minimum edge weights $2,3,4$, respectively, we'd return the path that had minimum edge $4$.
I'm trying to solve the problem as quickly as possible. I was considering using DFS to solve the problem, always starting at $s$ and trying every path to $t$, always keeping the minimum edge in memory, but I'm not sure if this is 1. fast and 2. practical.
In general, I'm just looking for an algorithm that can solve this problem as quickly as possible.