We generalize the ε-relaxation method of  for the single commodity, linear or separable convex cost network flow problem to network flow problems with positive gains. The method maintains ε-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 ε-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(ε0/ε̄), where ε0 and ε̄ denote, respectively, the initial and the final tolerance ε.
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