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Fast, Fair, and Efficient Flows in Networks

Author

Listed:
  • José R. Correa

    (School of Business, Universidad Adolfo Ibáñez, Av. Presidente Errázuriz 3485, Las Condes, Santiago, Chile)

  • Andreas S. Schulz

    (Sloan School of Management, Massachusetts Institute of Technology, E53-361, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139)

  • Nicolás E. Stier-Moses

    (Graduate School of Business, Columbia University, Uris Hall, Room 418, 3022 Broadway, New York, New York 10027)

Abstract

We study the problem of minimizing the maximum latency of flows in networks with congestion. We show that this problem is NP-hard, even when all arc latency functions are linear and there is a single source and sink. Still, an optimal flow and an equilibrium flow share a desirable property in this situation: All flow-carrying paths have the same length, i.e., these solutions are “fair,” which is in general not true for optimal flows in networks with nonlinear latency functions. In addition, the maximum latency of the Nash equilibrium, which can be computed efficiently, is within a constant factor of that of an optimal solution. That is, the so-called price of anarchy is bounded. In contrast, we present a family of instances with multiple sources and a single sink for which the price of anarchy is unbounded, even in networks with linear latencies. Furthermore, we show that an s - t -flow that is optimal with respect to the average latency objective is near-optimal for the maximum latency objective, and it is close to being fair. Conversely, the average latency of a flow minimizing the maximum latency is also within a constant factor of that of a flow minimizing the average latency.

Suggested Citation

  • José R. Correa & Andreas S. Schulz & Nicolás E. Stier-Moses, 2007. "Fast, Fair, and Efficient Flows in Networks," Operations Research, INFORMS, vol. 55(2), pages 215-225, April.
    • Handle: RePEc:inm:oropre:v:55:y:2007:i:2:p:215-225
    DOI: 10.1287/opre.1070.0383
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    References listed on IDEAS

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    1. Olaf Jahn & Rolf H. Möhring & Andreas S. Schulz & Nicolás E. Stier-Moses, 2005. "System-Optimal Routing of Traffic Flows with User Constraints in Networks with Congestion," Operations Research, INFORMS, vol. 53(4), pages 600-616, August.
    2. L. R. Ford & D. R. Fulkerson, 1958. "Constructing Maximal Dynamic Flows from Static Flows," Operations Research, INFORMS, vol. 6(3), pages 419-433, June.
    3. John J. Jarvis & H. Donald Ratliff, 1982. "Note---Some Equivalent Objectives for Dynamic Network Flow Problems," Management Science, INFORMS, vol. 28(1), pages 106-109, January.
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    Citations

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    Cited by:

    1. Parilina, Elena & Sedakov, Artem & Zaccour, Georges, 2017. "Price of anarchy in a linear-state stochastic dynamic game," European Journal of Operational Research, Elsevier, vol. 258(2), pages 790-800.
    2. Manxi Wu & Saurabh Amin & Asuman E. Ozdaglar, 2021. "Value of Information in Bayesian Routing Games," Operations Research, INFORMS, vol. 69(1), pages 148-163, January.
    3. Georgia Perakis & Wei Sun, 2014. "Efficiency Analysis of Cournot Competition in Service Industries with Congestion," Management Science, INFORMS, vol. 60(11), pages 2684-2700, November.
    4. Gaëtan Fournier & Marco Scarsini, 2014. "Hotelling Games on Networks: Efficiency of Equilibria," Post-Print halshs-00983085, HAL.
    5. Zhu, Feng & Ukkusuri, Satish V., 2017. "Efficient and fair system states in dynamic transportation networks," Transportation Research Part B: Methodological, Elsevier, vol. 104(C), pages 272-289.
    6. Feng, Zengzhe & Gao, Ziyou & Sun, Huijun, 2014. "Bounding the inefficiency of atomic splittable selfish traffic equilibria with elastic demands," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 63(C), pages 31-43.
    7. Bo Chen & Xujin Chen & Xiaodong Hu, 2010. "The price of atomic selfish ring routing," Journal of Combinatorial Optimization, Springer, vol. 19(3), pages 258-278, April.
    8. Riccardo Colini-Baldeschi & Roberto Cominetti & Panayotis Mertikopoulos & Marco Scarsini, 2020. "When Is Selfish Routing Bad? The Price of Anarchy in Light and Heavy Traffic," Operations Research, INFORMS, vol. 68(2), pages 411-434, March.
    9. Sanjiv Kapoor & Junghwan Shin, 2020. "Price of Anarchy in Networks with Heterogeneous Latency Functions," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 755-773, May.
    10. Angelelli, E. & Morandi, V. & Savelsbergh, M. & Speranza, M.G., 2021. "System optimal routing of traffic flows with user constraints using linear programming," European Journal of Operational Research, Elsevier, vol. 293(3), pages 863-879.
    11. Correa, José R. & Schulz, Andreas S. & Stier-Moses, Nicolás E., 2008. "A geometric approach to the price of anarchy in nonatomic congestion games," Games and Economic Behavior, Elsevier, vol. 64(2), pages 457-469, November.
    12. Bayram, Vedat & Tansel, Barbaros Ç. & Yaman, Hande, 2015. "Compromising system and user interests in shelter location and evacuation planning," Transportation Research Part B: Methodological, Elsevier, vol. 72(C), pages 146-163.
    13. Werth, T.L. & Holzhauser, M. & Krumke, S.O., 2014. "Atomic routing in a deterministic queuing model," Operations Research Perspectives, Elsevier, vol. 1(1), pages 18-41.

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