Concave Generalized Flows with Applications to Market Equilibria

We consider a nonlinear extension of the generalized network flow model, with the flow leaving an arc being an increasing concave function of the flow entering it, as proposed by Truemper and Shigeno. We give a polynomial time combinatorial algorithm for solving corresponding flow maximization problems, finding an $\varepsilon$-approximate solution in $O(m(m+\log n)\log(MUm/\varepsilon))$ arithmetic operations and value oracle queries, where $M$ and $U$ are upper bounds on simple parameters. This also gives a new algorithm for linear generalized flows, an efficient, purely scaling variant of the Fat-Path algorithm by Goldberg, Plot kin and Tardos, not using any cycle cancellations. We show that this general convex programming model serves as a common framework for several market equilibrium problems, including the linear Fisher market model and its various extensions. Our result immediately provides combinatorial algorithms for various extensions of these market models. This includes nonsymmetric Arrow-Debreu Nash bargaining, settling an open question by Vazirani [4].