Abstract. We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with link-costs and, for each pair {i, j} of nodes, an edgeconnectivity requirement rij. The goal is to find a minimum-cost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within 2[log2(r + 1)] of optimal, where r is the highest requirement value. In the course of proving the performance guarantee, we prove a combinatorial minmax approximate equality relating minimum-cost networks to maximum packings of certain kinds of cuts. As a consequence of the proof of this theorem, we obtain an approximation algorithm for optimally packing these cuts; we show that this algorithm has application to estimating the reliability of a probabilistic network.

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Philip Klein4Aj it Agrawalt R. Ravit Satish RaoJIn this paper, we prove the first approximate max-flow mincut theorem for general multicommodity flow. We use it to get approximation algorithms for minimum deletion of clauses of a P-CNFE formula, via minimization, and other problems. We also present approximation algorithms for chordalization of a graph and for register sufficiency, based on undirected and directed node separators .'A complementary result WM obtained by Johnson [12] in 1974; he rhowed that the maximum ri5c of a rstirfiable aubret of claurer could be approximated to within a small constant factor; indeed, for every formula, d but at most a constant fraction of the clauses could be satisfied simultaneously. In view of this latter result, it makes sense to focus on how many clauses must be deleted to achieve satisfiability. I26CH2925-6/90/0000/0726$01 .OO 0 1990 I EEE