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Concave Generalized Flows with Applications to Market Equilibria

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  • Laszlo A. Vegh

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.

Suggested Citation

  • Laszlo A. Vegh, 2011. "Concave Generalized Flows with Applications to Market Equilibria," Papers 1109.3893, arXiv.org, revised Apr 2012.
    • Handle: RePEc:arx:papers:1109.3893
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    1. Donald Goldfarb & Zhiying Jin & James B. Orlin, 1997. "Polynomial-Time Highest-Gain Augmenting Path Algorithms for the Generalized Circulation Problem," Mathematics of Operations Research, INFORMS, vol. 22(4), pages 793-802, November.
    2. Jain, Kamal & Vazirani, Vijay V., 2010. "Eisenberg-Gale markets: Algorithms and game-theoretic properties," Games and Economic Behavior, Elsevier, vol. 70(1), pages 84-106, September.
    3. Andrew V. Goldberg & Serge A. Plotkin & Éva Tardos, 1991. "Combinatorial Algorithms for the Generalized Circulation Problem," Mathematics of Operations Research, INFORMS, vol. 16(2), pages 351-381, May.
    4. Donald Goldfarb & Zhiying Jin, 1996. "A Faster Combinatorial Algorithm for the Generalized Circulation Problem," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 529-539, August.
    5. James B. Orlin, 1993. "A Faster Strongly Polynomial Minimum Cost Flow Algorithm," Operations Research, INFORMS, vol. 41(2), pages 338-350, April.
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