A polylogarithmic approximation algorithm for the group Steiner tree problem

Given a weighted graph with some subsets of verticcs called groups. the group Steiner tree problem is to find a minimum-weight subgraph which contains at least one vertex from each group. We give a randomized algorithm with a polylogarithmic approximation guarantee for the group Steiner tree problem. The previous best approximation guarantee was O(i(2)k(1/i)) in time O(n(i)k(2i)) (Charikar, Chekuri, Goel. and Guha). Our algorithm also improves existing approximation results fur network design problems with location-based constraints and for the symmetric generalized traveling salesman problem. (C) 2000 Academic Press.