  We give an algorithm to find a minimum cut in an edge-weighted directed graph
with $n$ vertices and $m$ edges in $\tilde O(n\cdot \max(m^{2/3}, n))$ time.
This improves on the 30 year old bound of $\tilde O(nm)$ obtained by Hao and
Orlin for this problem. Our main technique is to reduce the directed mincut
problem to $\tilde O(\min(n/m^{1/3}, \sqrt{n}))$ calls of {\em any} maxflow
subroutine. Using state-of-the-art maxflow algorithms, this yields the above
running time. Our techniques also yield fast {\em approximation} algorithms for
finding minimum cuts in directed graphs. For both edge and vertex weighted
graphs, we give $(1+\epsilon)$-approximation algorithms that run in $\tilde
O(n^2 / \epsilon^2)$ time.
