Abstract
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).
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References
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)
Beckmann, M., McGuire, C.B., Winsten, C.B.: Studies in the Economics of Transportation. Yale University Press (1956)
Braess, D.: Über ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung 12, 258–268 (1968)
Chau, C.K., Sim, K.M.: The price of anarchy for non-atomic congestion games with symmetric cost maps and elastic demands. Operations Research Letters 31(5), 327–335 (2003)
Correa, J.R., Schulz, A.S., Stier Moses, N.E.: Computational complexity, fairness, and the price of anarchy of the maximum latency problem. In: Bienstock, D., Nemhauser, G.L. (eds.) IPCO 2004. LNCS, vol. 3064, pp. 59–73. Springer, Heidelberg (2004)
Correa, J.R., Schulz, A.S., Stier Moses, N.E.: Selfish routing in capacitated networks. Mathematics of Operations Research 29(4), 961–976 (2004)
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)
Devanur, N., Garg, N., Khandekar, R., Pandit, V., Saberi, A.: Price of anarchy, locality gap, and a network service provider game (2003) (Unpublished manuscript)
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)
Feldmann, R., Gairing, M., Lücking, T., Monien, B., Rode, M.: Selfish routing in non-cooperative networks: A survey. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 21–45. Springer, Heidelberg (2003)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)
Johari, R., Tsitsiklis, J.N.: Efficiency loss in a network resource allocation game. Mathematics of Operations Research 29(3), 407–435 (2004)
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)
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)
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)
Perakis, G.: The price of anarchy when costs are non-separable and asymmetric. In: Bienstock, D., Nemhauser, G.L. (eds.) IPCO 2004. LNCS, vol. 3064, pp. 46–58. Springer, Heidelberg (2004)
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)
Roughgarden, T.: The price of anarchy is independent of the network topology. Journal of Computer and System Sciences 67(2), 341–364 (2003)
Roughgarden, T.: The maximum latency of selfish routing. In: Proceedings of the 15th Annual Symposium on Discrete Algorithms (SODA), pp. 973–974 (2004)
Roughgarden, T.: Selfish Routing and the Price of Anarchy. MIT Press, Cambridge (2005)
Roughgarden, T., Tardos, É.: How bad is selfish routing? Journal of the ACM 49(2), 236–259 (2002)
Roughgarden, T., Tardos, É.: Bounding the inefficiency of equilibria in nonatomic congestion games. Games and Economic Behavior 47(2), 389–403 (2004)
Smith, M.J.: The existence, uniqueness and stability of traffic equilibria. Transportation Research 13B, 295–304 (1979)
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)
Weitz, D.: The price of anarchy (2001) (Unpublished manuscript)
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Lin, H., Roughgarden, T., Tardos, É., Walkover, A. (2005). Braess’s Paradox, Fibonacci Numbers, and Exponential Inapproximability. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds) Automata, Languages and Programming. ICALP 2005. Lecture Notes in Computer Science, vol 3580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11523468_41
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DOI: https://doi.org/10.1007/11523468_41
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