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 /spl gamma/(e). If f(e) units of flow enter arc e, then f(e)/spl gamma/(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/spl tilde/(m/sup 2/(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/sup (4/3/-/spl epsiv/) and B is not extremely large, then our bound improves the previous best bound O(m/sup 1.5/n/sup 2/log 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+/spl xi/ from the optimum can be computed in O/spl tilde/(m/sup 2/n+min{m/sup 2/n, m(m+nloglog B)}log(1//spl xi/)) time. This improves the previous bounds on the approximate generalized flow problem.<
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