Compute a max-flow from a min-cut We know that computing a maximum flow resp. a minimum cut of a network with capacities
is equivalent; cf. the max-flow min-cut theorem.

We have (more or less efficient) algorithms for computing maximum flows, and computing a minimum cut given a maximum flow is neither hard nor expensive, either.

But what about the reverse? Given a minimum cut, how can we determine a maximum flow?
Without solving Max-Flow from scratch, of course, and preferably faster than that, too.

Some thoughts:


From the minimum cut, we know the maximum flow value. I don't see how this information helps the standard approaches augmenting-path and push-relabel, although adapting the latter seems slightly more plausible.
We can not use the minimum cut to split the network in two parts and recurse since that won't shrink the problem in the worst case (if one partition is a singleton); also we would not have a minimum cut of the smaller instances.
Does knowing the value of the maximum flow speed up solving the Max-Flow LP, maybe via the complementary slackness conditions?
In the worst case, the minimum cut itself doesn't convey much information about the maximum flow. Consider a graph $G=(V,E)$ in which the minimum $s,t$-cut has value $w$. If I extend $G$ by adding a new vertex $s'$ and an edge $(s',s)$ with weight $w$, a minimum $s',t$-cut in the new graph consists of just the edge $(s', s)$ but that doesn't give any information about how to get $w$ units of flow from $s$ to $t$.

Effectively, the minimum cut tells you the value of the flow, but not how to achieve that flow. This means that knowing the minimum cut can speed up finding the flow by at most a logarithmic factor, since we could do binary search to find the value of the cut.