An O(log2k)-Approximation Algorithm for the k-Vertex Connected Spanning Subgraph Problem

We present an $O(\log^2{k})$-approximation algorithm for the problem of finding a $k$-vertex connected spanning subgraph of minimum cost, where $n$ is the number of vertices in an input graph, and $k$ is a connectivity requirement. Our algorithm is the first that achieves a polylogarithmic approximation ratio for all values of $k$ and $n$, and it works for both directed and undirected graphs. As in previous works, we use the Frank--Tardos algorithm for finding $k$-outconnected subgraphs as a subroutine. However, with our structural lemmas, we are able to show that we need only partial solutions returned by the Frank--Tardos algorithm; thus, we can avoid paying the whole cost of an optimal solution every time the algorithm is applied.