A epsilon-Relaxation Method for Generalized Separable Convex Cost Network Flow Problems

We propose an extension of the e-relaxation method to generalized network flow problems with separable convex cost. The method maintains e-complementary slackness satisfied at all iterations and adjusts the arc flows and the node prices so to satisfy flow conservation upon termination. Each iteration of the method involves either a price change at a node or a flow change at an arc or a flow change around a simple cycle. Complexity bounds for the method are derived. For one implementation employing e-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(e0/e−), where e0 and e− denote, respectively, the initial and the final e.