An Ε-relaxation Method for Separable Convex Cost Generalized Network Flow Problems1

We generalize the -relaxation method of [BPT97a] for the single commodity, linear or separable convex cost network flow problem to network flow problems with positive gains. The method maintains -complementary slackness at all iterations and adjusts the arc flows and the node prices so as to satisfy flow conservation upon termination. Each iteration of the method involves either a price change on a node or a flow change along an arc or a flow change along a simple cycle. Complexity bounds for the method are derived. For one implementation employing -scaling, the bound is polynomial in the number of nodes N , the number of arcs A, a certain constant Γ depending on the arc gains, and ln( 0/ ), where 0 and denote, respectively, the initial and the final tolerance .