Let's have a directed graph $G=(V,E)$. If I pick some vertex to be the source $s$ and the some other vertex to be the sink $t$, I can calculate the max flow allowable on the graph from $s$ to $t$ using the known max flow/min cut algorithms.
Is there an analogous argument to finding a few related things: 1) the cut that minimizes the quantity min(s->t max flow, t->s max flow)? This would seem to me to be just two applications of max flow/min cut, and taking the smaller one; is this more subtle? 2) the cut that minimizes the sum of the s->t max flow and the t->s max flow? Is there also some efficient way of doing this, or does this turn into some hard joint optimization problem?
Many thanks!