An approximation algorithm for the Group Steiner Problem

The input in the Group-Steiner Problem consists of an undirected connected graph with a cost function p(e) over the edges and a collection of subsets of vertices {gi} Each subset gi is called a group and the vertices in Ugi are called terminals. The goal is to find a minimum cost tree that contains at least one terminal from every group. We give the first combinatorial polylogarith-mic ratio approximation for the problem on trees. Let m denote the number of groups and S denote the number of terminals. The approximation ratio of our algorithm is O(logS · log m/log log S) = O(log²n/loglogn). This is an improvement by a φ(log log n) factor over the previously best known ap proximation ratio for the Group Steiner Problem on trees [GKR98]. Our result carries over to the Group Steiner Problem on general graphs and to the Covering Steiner Problem. Garg et al. [GKR98] presented a reduction of the Group Steiner Problem on general graphs to trees. Their reduction employs Bar-tal's [B98] approximation of graph metrics by tree metrics. Our algorithm on trees implies an approximation algorithm of ratio O(logS · logm · logn · log log n/ log log S) = O(log³ n) for the Group Steiner Problem on general graphs. The previously best known approximation ratio for this problem on general graphs, as a function of n, is O(log³n · log logn) [GKR98]. Our algorithm in conjunction with ideas of [EKS01] gives an O(logS · logm· logn· log logn/ log log S) = O(log³ n)-approximation ratio for the more general Covering Steiner Problem, improving the best known approximation ratio (as a function of n) for the Covering Steiner Problem by a Φ(loglogn) factor.