Approximating Directed Connectivity in Almost-Linear Time

We present randomized algorithms that compute $(1+\epsilon)$-approximate minimum global edge and vertex cuts in weighted directed graphs in $O(\log^4(n) / \epsilon)$ and $O(\log^5(n)/\epsilon)$ single-commodity flows, respectively. With the almost-linear time flow algorithm of [CKL+22], this gives almost linear time approximation schemes for edge and vertex connectivity. By setting $\epsilon$ appropriately, this also gives faster exact algorithms for small vertex connectivity. At the heart of these algorithms is a divide-and-conquer technique called "shrink-wrapping" for a certain well-conditioned rooted Steiner connectivity problem. Loosely speaking, for a root $r$ and a set of terminals, shrink-wrapping uses flow to certify the connectivity from a root $r$ to some of the terminals, and for the remaining uncertified terminals, generates an $r$-cut where the sink component both (a) contains the sink component of the minimum $(r,t)$-cut for each uncertified terminal $t$ and (b) has size proportional to the number of uncertified terminals. This yields a divide-and-conquer scheme over the terminals where we can divide the set of terminals and compute their respective minimum $r$-cuts in smaller, contracted subgraphs.