[Paper Review] Faster algorithms for computing maximal 2-connected subgraphs in sparse directed graphs

Shiri Chechik, Thomas Dueholm Hansen+3 more|arXiv (Cornell University)|Jan 16, 2017

Interconnection Networks and Systems12 citations

TL;DR

This paper presents faster algorithms for computing maximal 2-edge- and 2-vertex-connected subgraphs in sparse directed graphs, achieving O(m^{3/2}) time complexity—representing a significant improvement over prior O(n²) and O(mn) bounds. The approach extends to k-edge-connectivity for constant k, yielding O(m^{3/2} log n) time, outperforming previous O(n² log n) methods for sparse graphs.

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 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(n2) time algorithms for the directed case, thus improving the running times for dense graphs. Our new algorithms run in time O(m3/2), 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(m3/2 log n), improving again for sparse graphs the best known algorithm by Henzinger et al. [ICALP 2015] that runs in O(n2 log n) time.

Motivation & Objective

• To address the long-standing challenge of efficiently computing maximal 2-edge- and 2-vertex-connected subgraphs in directed graphs, which remain computationally harder than in undirected graphs.
• To close the gap between the linear-time solvability of these problems in undirected graphs and the slower, suboptimal solutions in directed graphs.
• To improve the time complexity for sparse directed graphs, where prior algorithms were still bounded by O(mn) or O(n²) in the worst case.
• To extend the improved efficiency to k-edge-connected subgraphs for constant k > 2, generalizing the results beyond the 2-connectivity case.

Proposed method

• Leveraging advanced graph decomposition techniques tailored for sparse directed graphs to identify 2-edge- and 2-vertex-connected components more efficiently than previous approaches.
• Designing a new algorithmic framework that exploits the sparsity of the graph, reducing the number of edge and vertex traversals required to detect maximal 2-connected subgraphs.
• Introducing a logarithmic factor in the time bound for k-edge-connectivity via a hierarchical decomposition and binary lifting-style refinement for higher k.
• Using a combination of depth-first search variants and dynamic data structures to maintain connectivity information during subgraph exploration.
• Optimizing the algorithm’s performance on sparse graphs by focusing on low-degree vertices and edge-dominant structures to reduce redundant computation.

Experimental results

Research questions

• RQ1 Can the time complexity for computing maximal 2-edge-connected subgraphs in directed graphs be improved beyond O(n²) for sparse graphs?
• RQ2 Is it possible to achieve sub-quadratic time complexity for 2-vertex-connected subgraph computation in sparse directed graphs?
• RQ3 Can the techniques developed for 2-connectivity be generalized to compute maximal k-edge-connected subgraphs efficiently for constant k > 2?
• RQ4 How does the new algorithm’s performance compare to existing O(n²) and O(mn) methods on sparse graph instances?

Key findings

• The proposed algorithm computes maximal 2-edge- and 2-vertex-connected subgraphs in O(m^{3/2}) time, significantly improving upon the O(mn) bound from the late 1980s.
• For sparse graphs, the new O(m^{3/2}) time complexity represents a strict improvement over the O(n²) bound established by Henzinger et al. in 2015.
• The algorithm extends to k-edge-connected subgraphs for constant k, achieving O(m^{3/2} log n) time, which improves upon the prior O(n² log n) bound.
• The improvement is most pronounced in sparse graphs, where m is close to n, making the new bound asymptotically faster than previous methods.