Braess’s Paradox, Fibonacci Numbers, and Exponential Inapproximability
We give the first analyses in multicommodity networks of both the worst-case severity of Braess’s Paradox and the price of anarchy of selfish routing with respect to the maximum latency. Our first main result is a construction of an infinite family of two-commodity networks, related to the Fibonacci numbers, in which both of these quantities grow exponentially with the size of the network. This construction has wide implications, and demonstrates that numerous existing analyses of selfish routing in single-commodity networks have no analogues in multicommodity networks, even in those with only two commodities. This dichotomy between single- and two-commodity networks is arguably quite unexpected, given the negligible dependence on the number of commodities of previous work on selfish routing.
Our second main result is an exponential upper bound on the worst-possible severity of Braess’s Paradox and on the price of anarchy for the maximum latency, which essentially matches the lower bound when the number of commodities is constant.
Finally, we use our family of two-commodity networks to exhibit a natural network design problem with intrinsically exponential (in)approximability: while there is a polynomial-time algorithm with an exponential approximation ratio, subexponential approximation is unachievable in polynomial time (assuming P ≠ NP).
KeywordsNash Equilibrium Latency Function Average Latency Network Design Problem Maximum Latency
Unable to display preview. Download preview PDF.
- 1.Anshelevich, E., Dasgupta, A., Tardos, É., Wexler, T.: Near-optimal network design with selfish agents. In: Proceedings of the 35th Annual ACM Symposium on the Theory of Computing (STOC), pp. 511–520 (2003)Google Scholar
- 2.Beckmann, M., McGuire, C.B., Winsten, C.B.: Studies in the Economics of Transportation. Yale University Press (1956)Google Scholar
- 7.Czumaj, A.: Selfish routing on the Internet. In: Leung, J. (ed.) Handbook of Scheduling: Algorithms, Models, and Performance Analysis, ch. 42, CRC Press, Boca Raton (2004)Google Scholar
- 8.Devanur, N., Garg, N., Khandekar, R., Pandit, V., Saberi, A.: Price of anarchy, locality gap, and a network service provider game (2003) (Unpublished manuscript)Google Scholar
- 9.Fabrikant, A., Luthra, A., Maneva, E., Papadimitriou, C.H., Shenker, S.J.: On a network creation game. In: Proceedings of the 22nd ACM Symposium on Principles of Distributed Computing (PODC), pp. 347–351 (2003)Google Scholar
- 13.Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. In: Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science (STACS), pp. 404–413 (1999)Google Scholar
- 14.Lin, H., Roughgarden, T., Tardos, É.: A stronger bound on braess’s paradox. In: Proceedings of the 15th Annual Symposium on Discrete Algorithms (SODA), pp. 333–334 (2004)Google Scholar
- 15.Papadimitriou, C.H.: Algorithms, games, and the Internet. In: Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing (STOC), pp. 749–753 (2001)Google Scholar
- 17.Roughgarden, T.: Designing networks for selfish users is hard. In: Proceedings of the 42nd Annual Symposium on Foundations of Computer Science, pp. 472–481 (2001)Google Scholar
- 19.Roughgarden, T.: The maximum latency of selfish routing. In: Proceedings of the 15th Annual Symposium on Discrete Algorithms (SODA), pp. 973–974 (2004)Google Scholar
- 20.Roughgarden, T.: Selfish Routing and the Price of Anarchy. MIT Press, Cambridge (2005)Google Scholar
- 23.Smith, M.J.: The existence, uniqueness and stability of traffic equilibria. Transportation Research 13B, 295–304 (1979)Google Scholar
- 24.Vetta, A.: Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions. In: Proceedings of the 43rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 416–425 (2002)Google Scholar
- 25.Weitz, D.: The price of anarchy (2001) (Unpublished manuscript)Google Scholar