Abstract
Let G be a strongly connected directed graph. We consider the problem of computing the smallest strongly connected spanning subgraph of G that maintains the pairwise 2-vertex-connectivity of G, i.e., the 2-vertex-connected blocks of G (2VC-B). We provide linear-time approximation algorithms for this problem that achieve an approximation ratio of 6. Based on these algorithms, we show how to approximate, in linear time, within a factor of 6 the smallest strongly connected spanning subgraph of G that maintains respectively: both the 2-vertex-connected blocks and the 2-vertex-connected components of G (2VC-B-C); all the 2-connectivity relations of G (2C), i.e., both the 2-vertex- and the 2-edge-connected components and blocks. Moreover, we provide heuristics that improve the size of the computed subgraphs in practice, and conduct a thorough experimental study to assess their merits in practical scenarios.
G.F. Italiano and N. Parotsidis—Partially supported by MIUR under Project AMANDA.
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Notes
- 1.
Throughout, we use consistently the term bridge to refer to a bridge of a flow graph G(s) and the term strong bridge to refer to a strong bridge in the original graph G.
- 2.
This follows from the fact that in the sparse subgraph the k vertices in blocks must have indegree at least two, while the remaining \(n-k\) vertices must have indegree at least one, since we seek for a strongly connected spanning subgraph.
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Georgiadis, L., Italiano, G.F., Karanasiou, A., Papadopoulos, C., Parotsidis, N. (2016). Sparse Subgraphs for 2-Connectivity in Directed Graphs. In: Goldberg, A., Kulikov, A. (eds) Experimental Algorithms. SEA 2016. Lecture Notes in Computer Science(), vol 9685. Springer, Cham. https://doi.org/10.1007/978-3-319-38851-9_11
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