Faster Algorithms for the Generalized Network Flow Problem

We consider the generalized network flow problem. Each arc e in the network has a gain factor y (e ) . If f ( e ) units of flow enter arc e , then f ( e ) r ( e ) units arrive at the other end of e. The generalized network flow problem is to maximize the net flow into one specific node, the sink. -We give an algorithm which solves this problem in O(m2 (m + n log log B ) log B ) time, where B is the largest integer used to represent the gain factors, the capacities, and the initial supplies at the nodes. If m is O(n4/3-‘) and B is not extremely large, then our bound improves the previous best bound O(m’.5n2 log B ) given by Vaidya 191. Our algorithm is an approximation scheme which in each iteration reduces by a constant factor the difference between the current net flow into the sink and the optimal one. The solution which is within a fact_or of 1 + .$ from the optimum can be computed in O(m2n + min{m2n, m(m + nloglogB)} log(l/J)) time. This improves the previous bounds on the approximate generalized flow problem.