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Network Characterizations for Excluding Braess’s Paradox

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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.

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  1. Edges and arcs are collectively called links. An undirected link is an edge, and a directed link is an arc.

  2. We emphasize again that all paths in this paper are simple. They are all acyclic.

  3. Milchtaich’s proof [20] concerned only undirected graphs. The directed case is simply an immediate corollary.


  1. 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)

  2. Beckmann, M., McGuire, B., Winsten, C.B: Studies in the Economics of Transportation. Technical report (1956)

  3. Bell, M.G.H., Iida, Y.: Transportation Network Analysis (1997)

  4. Braess, D.: Über ein paradoxon aus der verkehrsplanung. Unternehmensforschung 12(1), 258–268 (1968)

    MathSciNet  MATH  Google Scholar 

  5. Cenciarelli, P., Gorla, D., Salvo, I.: Graph theoretic investigations on inefficiencies in network models. Discret. Math. (2016). arXiv:1603.01983

  6. Cohen, J.E., Horowitz, P.: Paradoxical behaviour of mechanical and electrical networks. Nature 352, 699–701 (1991)

    Article  Google Scholar 

  7. 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)

  8. 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)

  9. Diestel, R.: Graph Theory. Electronic Library of Mathematics. Springer (2000)

  10. Epstein, A., Feldman, M., Mansour, Y.: Efficient graph topologies in network routing games. Games Econ. Behav. 66(1), 115–125 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theor. Comput. Sci. 10(2), 111–121 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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

  14. Holzman, R., Monderer, D.: Strong equilibrium in network congestion games: Increasing versus decreasing costs. Int. J. Game Theory 1–20 (2014)

  15. Holzman, R., Nissan Law yon (Lev-tov): Network structure and strong equilibrium in route selection games. Math. Soc. Sci. 46(2), 193–205 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. Comput. Sci. Rev. 3, 65–69 (2009)

    Article  MATH  Google Scholar 

  17. 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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Milchtaich, I.: Topological conditions for uniqueness of equilibrium in networks. Math. Oper. Res. 30(1), 225–244 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

  20. Milchtaich, I.: Network topology and the efficiency of equilibrium. Games Econ. Behav. 57(2), 321–346 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. John, D.: Murchland. Braess’s paradox of traffic flow. Transp. Res. 4(4), 391–394 (1970)

    Article  Google Scholar 

  22. 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Roughgarden, T., Tardos, É.: How bad is selfish routing? J. ACM 49(2), 236–259 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schrijver, A.: Finding k disjoint paths in a directed planar graph. SIAM J. Comput. 23(4), 780–788 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Steinberg, R., Zangwill, W.I.: The prevalence of Braess’ paradox. Transp. Sci. 17(3), 301–318 (1983)

    Article  Google Scholar 

  26. Tutte, W.T.: Graph Theory. Electronic Library of Mathematics. China Machine Press (2004)

  27. Valdes, J., Tarjan, R.E., Lawler, E.L.: The recognition of series parallel digraphs. SIAM J. Comput. 11(2), 298–313 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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)

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The authors are indebted to anonymous referees for their invaluable comments and suggestions which have greatly improved the presentation of this paper.

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Correspondence to Xujin Chen.

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Research supported in part by NNSF of China under Grant No. 11531014 and 11222109.

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Chen, X., Diao, Z. & Hu, X. Network Characterizations for Excluding Braess’s Paradox. Theory Comput Syst 59, 747–780 (2016).

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