Faster Algorithms for Rooted Connectivity in Directed Graphs

TL;DR

This paper introduces faster randomized algorithms for computing rooted and global connectivity in directed graphs, improving efficiency especially in high-connectivity scenarios and extending to vertex connectivity with approximation guarantees.

Contribution

It presents the first sub-cubic algorithm for rooted edge connectivity in dense graphs and extends techniques to approximate and exact vertex connectivity in directed graphs.

Findings

New randomized $ ilde{O}(n^2)$ algorithm for rooted edge connectivity with small capacities.

Approximation algorithms for rooted and global vertex connectivity with near-quadratic time complexity.

Improved bounds for high-connectivity regimes in directed graphs.

Abstract

We consider the fundamental problems of determining the rooted and global edge and vertex connectivities (and computing the corresponding cuts) in directed graphs. For rooted (and hence also global) edge connectivity with small integer capacities we give a new randomized Monte Carlo algorithm that runs in time $\tilde{O}(n^2)$. For rooted edge connectivity this is the first algorithm to improve on the $\Omega(n^3)$ time bound in the dense-graph high-connectivity regime. Our result relies on a simple combination of sampling coupled with sparsification that appears new, and could lead to further tradeoffs for directed graph connectivity problems. We extend the edge connectivity ideas to rooted and global vertex connectivity in directed graphs. We obtain a $(1 + \epsilon)$-approximation for rooted vertex connectivity in $\tilde{O}(nW/\epsilon)$ time where $W$ is the total vertex weight (assuming integral vertex weights); in particular this yields an $\tilde{O}(n^2/\epsilon)$ time randomized algorithm for unweighted graphs. This translates to a $\tilde{O}(\kappa nW)$ time exact algorithm where $\kappa$ is the rooted connectivity. We build on this to obtain similar bounds for global vertex connectivity. Our results complement the known results for these problems in the low connectivity regime due to work of Gabow [9] for edge connectivity from 1991, and the very recent work of Nanongkai et al. [24] and Forster et al. [7] for vertex connectivity.