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Computational Complexity, Fairness, and the Price of Anarchy of the Maximum Latency Problem

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  • Correa, Jose R.
  • Schulz, Andreas S.
  • Stier Moses, Nicolas E.

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, one can prove that 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 the optimal flow 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 that shows that the price of anarchy is unbounded for instances with multiple sources and a single sink, even in networks with linear latencies. Finally, 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 latenc

Suggested Citation

  • Correa, Jose R. & Schulz, Andreas S. & Stier Moses, Nicolas E., 2004. "Computational Complexity, Fairness, and the Price of Anarchy of the Maximum Latency Problem," Working papers 4447-03, Massachusetts Institute of Technology (MIT), Sloan School of Management.
    • Handle: RePEc:mit:sloanp:5051
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    File URL: http://hdl.handle.net/1721.1/5051
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    Cited by:

    1. Haoning Xi & Liu He & Yi Zhang & Zhen Wang, 2020. "Bounding the efficiency gain of differentiable road pricing for EVs and GVs to manage congestion and emissions," PLOS ONE, Public Library of Science, vol. 15(7), pages 1-36, July.
    2. Huibing Yin & Prashant Mehta & Sean Meyn & Uday Shanbhag, 2014. "On the Efficiency of Equilibria in Mean-Field Oscillator Games," Dynamic Games and Applications, Springer, vol. 4(2), pages 177-207, June.
    3. Georgia Perakis, 2007. "The “Price of Anarchy” Under Nonlinear and Asymmetric Costs," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 614-628, August.
    4. 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.
    5. 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.

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