Fast Approximations for Rooted Connectivity in Weighted Directed Graphs

We consider approximations for computing minimum weighted cuts in directed graphs. We consider both rooted and global minimum cuts, and both edge-cuts and vertex-cuts. For these problems we give randomized Monte Carlo algorithms that compute a $(1+\epsilon)$-approximate minimum cut in $\tilde{O}(n^2 / \epsilon^2)$ time. These results extend and build on recent work [4] that obtained exact algorithms with similar running times in directed graphs with small integer capacities.