Polynomial combinatorial algorithm for generalized minimum cost flow

We develop the first polynomial combinatorial algorithm for generalized minimum cost flow. Despite a rich history dating back to Kantorovich and Dantzig, until now, the only known way to solve the problem in polynomial-time was via ellipsoid or interior point methods. Flow-based polynomial algorithms were previously known only for the version of our problem without costs. We design the first flow-based polynomial algorithms for the version with costs. We can also find provably good solutions faster than optimal ones. We obtain the first strongly polynomial approximation schemes for generalized network flow with costs. Our techniques also extend to optimize linear programs with two variables per inequality. Polynomial combinatorial algorithms were previously developed for testing the feasibility of such linear programs. Until now, no such methods were known for the optimization version.