Improved approximation bounds for the group Steiner problem

Given a weighted graph 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 O(k/sup /spl epsiv//)-approximation algorithms for arbitrarily small values of /spl epsiv/>0, improving on the previously known O(k/sup 1/2 /)-approximation. Our techniques also solve the graph Steiner arborescence problem with an O(k/sup /spl epsiv//) approximation bound. These results are directly applicable to a practical problem in VLSI layout, namely the routing of nets with multi-port terminals. Our Java implementation is available on the Web.