Let $G$ be a finite directed graph, and let $s,t$ be two distinct vertices.
Problem $1(s,t)$. Find the maximum number of mutually edge-disjoint directed paths from $s$ to $t$. OK, I didn't think of it first; it is the standard unit capacity flow problem about which there is a vast literature with many sophisticated algorithms.
Problem 2. Let's say that a "general $\{s,t\}$ path" is either a directed path from $s$ to $t$ or a directed path from $t$ to $s$. How can we find the maximum number of mutually edge-disjoint general $\{s,t\}$ paths?
It is easy to make examples where the optimum for Problem 2 is less than the optimum for Problem $1(s,t)$ plus the optimum for Problem $1(t,s)$.
The same problem can be asked with "edge-disjoint" replaced by "internally vertex-disjoint", though it probably reduces to the edge-disjoint case by the same reduction as used for Problem 1.
