Abstract
Braess’s paradox exposes a counterintuitive phenomenon that when travelers selfishly choose their routes in a network, removing links can improve the overall network performance. Under the model of nonatomic selfish routing, we characterize the topologies of k-commodity undirected and directed networks in which Braess’s paradox never occurs. Our results strengthen Milchtaich’s series-parallel characterization (Milchtaich, Games Econom. Behav. 57(2), 321–346 (2006)) for the single-commodity undirected case.
Similar content being viewed by others
Notes
Edges and arcs are collectively called links. An undirected link is an edge, and a directed link is an arc.
We emphasize again that all paths in this paper are simple. They are all acyclic.
Milchtaich’s proof [20] concerned only undirected graphs. The directed case is simply an immediate corollary.
References
Azar, Y., Epstein, A.: The hardness of network design for unsplittable flow with selfish users. In: Erlebach, T., Persinao, G. (eds.) Approximation and Online Algorithms, Lecture Notes in Computer Science, vol. 3879, pp. 41–54. Springer Berlin Heidelberg (2006)
Beckmann, M., McGuire, B., Winsten, C.B: Studies in the Economics of Transportation. Technical report (1956)
Bell, M.G.H., Iida, Y.: Transportation Network Analysis (1997)
Braess, D.: Über ein paradoxon aus der verkehrsplanung. Unternehmensforschung 12(1), 258–268 (1968)
Cenciarelli, P., Gorla, D., Salvo, I.: Graph theoretic investigations on inefficiencies in network models. Discret. Math. (2016). arXiv:1603.01983
Cohen, J.E., Horowitz, P.: Paradoxical behaviour of mechanical and electrical networks. Nature 352, 699–701 (1991)
Cygan, M., Marx, D., Pilipczuk, M.: The planar directed k-vertex-disjoint paths problem is fixed-parameter tractable. In: IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), 2013, pp. 197–206. IEEE (2013)
Czumaj, A.: Selfish routing on the internet. In: Leung, J.Y.-T. (ed.) Handbook of Scheduling: Algorithms, Models, and Performance Analysis, chap. 42. CRC Press (2004)
Diestel, R.: Graph Theory. Electronic Library of Mathematics. Springer (2000)
Epstein, A., Feldman, M., Mansour, Y.: Efficient graph topologies in network routing games. Games Econ. Behav. 66(1), 115–125 (2009)
Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theor. Comput. Sci. 10(2), 111–121 (1980)
Fotakis, D., Kaporis, A.C., Lianeas, T., Spirakis, P.G.: On the hardness of network design for bottleneck routing games. Theor. Comput. Sci. 521, 107–122 (2014)
Fujishige, S., Goemans, M.X., Harks, T., Peis, B., Zenklusen, R.: Matroids are immune to Braess paradox. Computer Science and Game Theory (2015). arXiv:1504.07545v1
Holzman, R., Monderer, D.: Strong equilibrium in network congestion games: Increasing versus decreasing costs. Int. J. Game Theory 1–20 (2014)
Holzman, R., Nissan Law yon (Lev-tov): Network structure and strong equilibrium in route selection games. Math. Soc. Sci. 46(2), 193–205 (2003)
Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. Comput. Sci. Rev. 3, 65–69 (2009)
Lin, H., Roughgarden, T., Tardos, É., Walkover, A.: Stronger bounds on Braess’s paradox and the maximum latency of selfish routing. SIAM J. Discret. Math. 25(4), 1667–1686 (2011)
Milchtaich, I.: Topological conditions for uniqueness of equilibrium in networks. Math. Oper. Res. 30(1), 225–244 (2005)
Milchtaich, I.: The equilibrium existence problem in finite network congestion games. In: Spirakis, P., Mavronicolas, M., Kontogiannis, S. (eds.) Internet and Network Economics, Lecture Notes in Computer Science, vol. 4286, pp. 87–98. Springer Berlin Heidelberg (2006)
Milchtaich, I.: Network topology and the efficiency of equilibrium. Games Econ. Behav. 57(2), 321–346 (2006)
John, D.: Murchland. Braess’s paradox of traffic flow. Transp. Res. 4(4), 391–394 (1970)
Roughgarden, T.: On the severity of Braess’s paradox: Designing networks for selfish users is hard. J. Comput. Syst. Sci. 72(5), 922–953 (2006)
Roughgarden, T., Tardos, É.: How bad is selfish routing? J. ACM 49(2), 236–259 (2002)
Schrijver, A.: Finding k disjoint paths in a directed planar graph. SIAM J. Comput. 23(4), 780–788 (1994)
Steinberg, R., Zangwill, W.I.: The prevalence of Braess’ paradox. Transp. Sci. 17(3), 301–318 (1983)
Tutte, W.T.: Graph Theory. Electronic Library of Mathematics. China Machine Press (2004)
Valdes, J., Tarjan, R.E., Lawler, E.L.: The recognition of series parallel digraphs. SIAM J. Comput. 11(2), 298–313 (1982)
Wardrop, J.G.: Some theoretical aspects of road traffic research. In: Proceedings of the Institute of Civil Engineers, Part II, vol. 1, pp. 325–378 (1952)
Acknowledgments
The authors are indebted to anonymous referees for their invaluable comments and suggestions which have greatly improved the presentation of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by NNSF of China under Grant No. 11531014 and 11222109.
Rights and permissions
About this article
Cite this article
Chen, X., Diao, Z. & Hu, X. Network Characterizations for Excluding Braess’s Paradox. Theory Comput Syst 59, 747–780 (2016). https://doi.org/10.1007/s00224-016-9710-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-016-9710-4