Approximating multiroot 3-outconnected subgraphs

The multiroot k-outconnected subgraph problem is: given an undirected graph G with nonnegative costs on the edges, a vector of q root nodes R = (r₁, ..., r_q), and a vector K = (k₁, ..., k_q) of connectivity requirements (with k = max k_i), find a minimum-cost K-outconnected from R spanning subgraph (that is, for every i = 1, ..., q there are k_i internally vertex disjoint paths from r_i to any other node). The best known algorithm for this problem has approximation ratio 2q, where q can be as large as k-1. For no value of k≥2 a better approximation algorithm is known. We consider the case k_i∈{1, 2, 3} which may arise in practical networks. For this case we give a 10/3 -approximation algorithm, improving the best previously known 4-approximation. Our algorithm also implies an improvement for arbitrary k. In the case we have an initial graph which is 2-connected, our algorithm achieves approximation ratio 2.