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 if the input also specifies a subset S \subseteq V of terminals and the goal is to find a subgraph containing at least k terminals. Closely related to the k-2VC problem, and in fact a special case of it, is the k-2EC problem, in which the goal is to find a minimum-cost 2-edge-connected subgraph containing k vertices. The k-2EC problem was introduced by Lau et al., who also gave a poly-logarithmic approximation for it. No previous approximation algorithm was known for the more general k-2VC problem. We describe an O(\log n \log k) approximation for the k-2VC problem. 
Chandra Chekuri, Nitish Korula