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 {g(i)}. Each subset gi is called a group and the vertices in boolean ORg(i) 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 polylogarithmic 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 0(logS . log m/log log S) = 0(log(2)n/log log n). This is an improvement by a Theta(log log n) factor over the previously best known approximation 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 Bartal's [1398] approximation of graph metrics by tree metrics. Our algorithm on trees implies an approximation algorithm of ratio 0(logS . logm . log n . log log n/log log S) = 0(log(3)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(log(3)n log log n) [GKR98]. Our algorithm in conjunction with ideas of [EKS01] gives an O(log S.log m.log n. log log n/log log S) = O(log(3)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 Theta(log log n) factor.