Faster Algorithms for Computing Maximal 2-Connected Subgraphs in Sparse Directed Graphs

TL;DR

This paper introduces faster algorithms for finding maximal 2-connected subgraphs in sparse directed graphs, significantly improving efficiency over previous methods and extending to higher connectivity levels.

Contribution

The authors present new algorithms with improved time complexity for computing maximal 2-edge- and 2-vertex-connected subgraphs in directed graphs, especially optimized for sparse graphs.

Findings

Algorithms run in O(m^{3/2}) time for sparse graphs.

Extended algorithms compute maximal k-edge-connected subgraphs in O(m^{3/2} log n) time.

Improved upon previous algorithms with higher time complexities.

Abstract

Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected (resp., 2-vertex-connected) if the removal of any edge (resp., vertex) leaves the graph strongly connected. In this paper we present improved algorithms for computing the maximal 2-edge- and 2-vertex-connected subgraphs of a given directed graph. These problems were first studied more than 35 years ago, with $\widetilde{O}(mn)$ time algorithms for graphs with m edges and n vertices being known since the late 1980s. In contrast, the same problems for undirected graphs are known to be solvable in linear time. Henzinger et al. [ICALP 2015] recently introduced $O(n^2)$ time algorithms for the directed case, thus improving the running times for dense graphs. Our new algorithms run in time $O(m^{3/2})$, which further improves the running times for sparse graphs. The notion of 2-connectivity naturally generalizes to k-connectivity for $k>2$. For constant values of k, we extend one of our algorithms to compute the maximal k-edge-connected in time $O(m^{3/2} \log{n})$, improving again for sparse graphs the best known algorithm by Henzinger et al. [ICALP 2015] that runs in $O(n^2 \log n)$ time.