Approximating Multiroot 3-Outconnected Subgraphs

The multiroot k-outconnected subgraph problem is: given an undirected graph with nonnegative costs on the edges, a vector of q root nodes ~ R = (r 1 ; : : : ; r q ), and a vector ~ K = (k 1 ; : : : ; k q ) of connectivity requirements (with k = max k i ), find a minimum-cost subgraph such that for every i = 1; : : : ; q there are k i internally vertex disjoint paths between r i and any other node. For k 2 this problem is NP-hard. It generalizes the problem of finding a minimum cost k-vertex connected spanning subgraph. The best known algorithm for the multiroot problem has approximation ratio 2q, where q can be as large as k \Gamma 1. For a general instance of the problem, for no value of k 2 a better approximation algorithm is known. We consider the case of small requirements k i 2 f1; 2; 3g; these may arise in applications, as in practical networks the connectivity requirements are usually rather small. For this case we give an algorithm with approximation ratio 10 3 . This impr..