Approximating multiroot 3-outconnected subgraphs

Consider the following problem: Given an undirected graph with nonnegative edge costs and requirements k_u for every node u, find a minimum-cost subgraph that contains max{k_u, k_v} internally disjoint paths between every pair of nodes u, v. For k = max k_u ≥ 2, this problem is NP-hard. The best-known algorithm for it has an approximation ratio of 2(k - 1). For a general instance of the problem, for no value of k ≥ 2, a better approximation algorithm was known. We consider the case of small requirements k_u ∈ {1, 2, 3}; these may arise in applications, as, in practical networks, the connectivity requirements are usually rather small. For this case, we give an algorithm with an approximation ratio of 10/3. This improves the best previously known approximation ratio of 4. Our algorithm also implies an improvement for arbitrary k. In the case in which we have an initial graph which is 2-connected, our algorithm achieves an approximation ratio of 2.