Approximating the Smallest 2-Vertex Connected Spanning Subgraph of a Directed Graph

Abstract

We consider the problem of approximating the smallest 2-vertex connected spanning subgraph (2-VCSS) of a 2-vertex connected directed graph, and explore the efficiency of fast heuristics. First, we present a linear-time heuristic that gives a 3-approximation of the smallest 2-VCSS. Then we show that this heuristic can be combined with an algorithm of Cheriyan and Thurimella that achieves a (1 + 1/k)-approximation of the smallest k-VCSS. The combined algorithm preserves the 1.5 approximation guarantee of the Cheriyan-Thurimella algorithm for k = 2 and improves its running time from O(m ²) to $O(m\sqrt{n}+n^2)$, for a digraph with n vertices and m arcs. Finally, we present an experimental evaluation of the above algorithms for a variety of input data. The experimental results show that our linear-time heuristic achieves in practice a much better approximation ratio than 3, suggesting that a tighter analysis may be possible. Furthermore, the experiments show that the combined algorithm not only improves the running time of the Cheriyan-Thurimella algorithm, but it may also compute a smaller 2-VCSS.

Research funded by the John S. Latsis Public Benefit Foundation. The sole responsibility for the content of this paper lies with its authors.