Approximating Directed Connectivity in Almost-Linear Time

2025-11-28

AI Summary

This paper addresses the (1+ε)-approximation of global minimum edge and vertex cuts in weighted directed graphs. We propose a randomized divide-and-conquer algorithm based on single-commodity flows, whose core innovation is a “shrink-wrapping” graph contraction technique: guided by root Steiner connectivity, the algorithm recursively partitions and contracts the input graph into smaller subgraphs, enabling independent approximate cut computation on each contracted instance and avoiding redundant global computations. The algorithm requires only $O(log^4 n / varepsilon)$ single-commodity flow calls to compute a (1+ε)-approximate minimum edge cut and $O(log^5 n / varepsilon)$ calls for a (1+ε)-approximate minimum vertex cut, achieving nearly linear overall running time. This is the first subquadratic flow-call complexity for approximate minimum cuts in directed graphs. Moreover, our method accelerates exact vertex-connectivity verification for small values of k.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

We present randomized algorithms that compute $(1+ε)$-approximate minimum global edge and vertex cuts in weighted directed graphs in $O(log^4(n) / ε)$ and $O(log^5(n)/ε)$ 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 $ε$ 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.

Problem

• Approximates directed graph connectivity in near-linear time
• Develops algorithms for edge and vertex cut approximations
• Introduces shrink-wrapping for rooted Steiner connectivity problems

Innovation

• Randomized algorithms for approximate directed connectivity
• Shrink-wrapping technique for rooted Steiner connectivity
• Almost-linear time using flow computations