Given a weighted g r aph and a family of k disjoint groups of nodes, the Group Steiner Problem asks for a minimum-cost routing tree that contains at least one node from each group. We give polynomial-time Ok -approximation algorithms for arbitrarily small values of 0 , improving on the previously known Ok 1 2 -approximation. Our techniques also solve the graph Steiner arborescence p r oblem with an Ok approximation bound. These results are directly applicable to a practical problem in VLSI

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... t, namely the routing of nets with multi-port terminals. Our Java implementation is available on the Web. 1 The performance r atio is an upper bound on the ratio of a heuristic solution cost divided by the optimal solution cost.