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 γ(e). If f(e) units of flow enter arc e, then f(e)γ(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+nloglog 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(n⁽⁴3/-ε) and B is not extremely large, then our bound improves the previous best bound O(m^1.5n²log B) given by P.M. Vaidya (1989). 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 factor of 1+ξ from the optimum can be computed in O˜(m²n+min{m ²n, m(m+nloglog B)}log(1/ξ)) time. This improves the previous bounds on the approximate generalized flow problem