Min-Cost 2-Connected Subgraphs With k Terminals

In the k-2VC problem, we are given an undirected graph G with edge costs and an integer k; the goal is to find a minimum-cost 2-vertex-connected subgraph of G containing at least k vertices. A slightly more general version is obtained in which the input also specifies a subset S V of terminals and the goal is to find a subgraph containing at least k terminals. Closely related to the k-2VC problem is the k-2EC problem in which the goal is to find a minimum-cost 2-edge-connected subgraph containing k vertices. We provide O(logn· logk) approximations for both the k-2VC and k-2EC problems. The k-2EC problem was first defined in (22), who claimed1 an O(log3 k)-approximation algorithm for it. No previous approximation algorithm was known for the k-2VC problem.