Simple Generalized Maximum Flow Algorithms

We introduce a gain-scaling technique for the generalized maximum ow problem. Using this technique, we present three simple and intuitive polynomial-time combinatorial algorithms for the problem. Truemper's augmenting path algorithm is one of the simplest combinatorial algorithms for the problem, but runs in exponential-time. Our rst algorithm is a polynomial-time variant of Truemper's algorithm. Our second algorithm is an adaption of Goldberg and Tarjan's pre owpush algorithm. It is the rst polynomial-time pre ow-push algorithm in generalized networks. Our third algorithm is a variant of the Fat-Path capacity-scaling algorithm. It is much simpler than Radzik's variant and matches the best known complexity for the problem. We discuss practical improvements in implementation.