Approximating the smallest 2-vertex connected spanning subgraph of a directed graph

We consider the problem of approximating the smallest 2-vertex connected spanning subgraph (2VCSS) of a 2-vertex connected directed graph, and provide new efficient algorithms. We provide two linear-time algorithms, the first based on a linear-time test for 2-vertex connectivity and divergent spanning trees, and the second based on low-high orders, that correspondingly give 3- and 2-approximations. Then we show that these linear-time algorithms can be combined with an algorithm of Cheriyan and Thurimella that achieves a 3/2-approximation. The combined algorithms preserve the 3/2 approximation guarantee of the Cheriyan-Thurimella algorithm and improve its running time from O(m2) to O(mn+n2), for a digraph with n vertices and m edges. Finally, we present an experimental evaluation of the above algorithms for a variety of input data. The experimental results show that our linear-time algorithms perform very well in practice. Furthermore, the experiments show that the combined algorithms not only improve the running time of the Cheriyan-Thurimella algorithm, but it may also compute a better solution.