An o(log ² k)-approximation algorithm for the k-vertex connected spanning subgraph problem

Abstract

We present an O(log n• log k)-approximation algorithm for the problem of finding k-vertex connected spanning subgraph of minimum cost, where n is the number of vertices in the input graph, and k is the connectivity requirement. Our algorithm works for both directed and undirected graphs. The best known approximation guarantees for these problems are O(ln k• min{√k,n/n-k ln k}) by Kortsarz and Nutov, and O(ln{k}) in the case of undirected graphs where n≥ 6k2 by Cheriyan, Vempala, and Vetta. Our algorithm is the first that has a polylogarithmic guarantee for all values of k. Combining our algorithm with the algorithm of Kortsarz and Nutov in case of small k, e.g., k<n/2, we have an O(log2 k)-approximation algorithm. As in previous work, we use the Frank-Tardos algorithm for finding k-outconnected subgraphs as a subroutine. However, with a structural lemmas that we proved, 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 the optimal solution every time the algorithm is applied.