Concave Generalized Flows with Applications to Market Equilibria

TL;DR

This paper introduces a polynomial-time combinatorial algorithm for solving a nonlinear extension of generalized network flows with concave arc functions, applicable to market equilibrium problems and Nash bargaining, improving computational efficiency and extending existing models.

Contribution

It provides the first polynomial-time combinatorial algorithm for concave generalized flows and extends market equilibrium models to more general settings.

Findings

Efficient epsilon-approximate solutions in polynomial time

New algorithm for linear generalized flows without cycle cancellations

Extension of market models to broader nonlinear settings

Abstract

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 epsilon-approximate solution in O(m(m+log n)log(MUm/epsilon)) 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, Plotkin 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 extends these market models to more general settings. We also obtain a combinatorial algorithm for nonsymmetric Arrow-Debreu Nash bargaining, settling an open question by Vazirani.