Concave Generalized Flows with Applications to Market Equilibria

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 [39] and Shigeno [35]. We give a polynomial time combinatorial algorithm for solving corresponding flow maximization problems, finding an ε-approximate solution in O(m(m + log n) log(M U m/ε)) 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 [12], 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 [41].

A strongly polynomial algorithm for generalized flow maximization

“…It is a cleaner and more efficient framework than similar ones in [11] and [34]; we believe this is the "real" condition a scaling type algorithm has to maintain.…”

“…This was followed by a multitude of further combinatorial algorithms e.g. [3,11,13,31,6,12,14,35,26,27,34]; a central motivation of this line of research was to develop a strongly polynomial algorithm. The algorithms of Cohen and Megiddo [3], Wayne [35], and Restrepo and Williamson [27] present fully polynomial time approximation schemes, that is, for every ε > 0, they can find a solution within ε from the optimum value in running time polynomial in n, m and log(1/ε).…”

“…The analogue of the scaling method was an important component of the Fat-Path algorithm of [9]; the algorithm of Goldfarb, Jin and Orlin [13] and the one in [34] also use this technique. The notion of "abundant arcs" can be easily extended to these frameworks: if an arc e carries a "large" amount of flow as compared to ∆, then it must be tight in every dual optimal solution, and hence can be contracted.…”

“…[9,11,34]) is sending ∆ units of relabeled flow on a tight path; we shall call this a path augmentation. In all previous approaches, the scaling factor ∆ remained fixed for a number of path augmentations, and reduces by a substantial amount (by at least a factor of two) for the next ∆-phase.…”

“…This is achieved by doing the exact opposite of [9,11,34] that use the natural analogue of the scaling technique for minimum cost circulations.…”