Greedy Selfish Network Creation

  • Pascal Lenzner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7695)

Abstract

We introduce and analyze greedy equilibria (GE) for the well-known model of selfish network creation by Fabrikant et al. [PODC’03]. GE are interesting for two reasons: (1) they model outcomes found by agents which prefer smooth adaptations over radical strategy-changes, (2) GE are outcomes found by agents which do not have enough computational resources to play optimally. In the model of Fabrikant et al. agents correspond to Internet Service Providers which buy network links to improve their quality of network usage. It is known that computing a best response in this model is NP-hard. Hence, poly-time agents are likely not to play optimally. But how good are networks created by such agents? We answer this question for very simple agents. Quite surprisingly, naive greedy play suffices to create remarkably stable networks. Specifically, we show that in the Sum version, where agents attempt to minimize their average distance to all other agents, GE capture Nash equilibria (NE) on trees and that any GE is in 3-approximate NE on general networks. For the latter we also provide a lower bound of \(\tfrac{3}{2}\) on the approximation ratio. For the Max version, where agents attempt to minimize their maximum distance, we show that any GE-star is in 2-approximate NE and any GE-tree having larger diameter is in \(\tfrac{6}{5}\)-approximate NE. Both bounds are tight. We contrast these positive results by providing a linear lower bound on the approximation ratio for the Max version on general networks in GE. This result implies a locality gap of Ω(n) for the metric min-max facility location problem, where n is the number of clients.

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References

  1. 1.
    Albers, S., Eilts, S., Even-Dar, E., Mansour, Y., Roditty, L.: On nash equilibria for a network creation game. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithm, SODA 2006, pp. 89–98. ACM, New York (2006)Google Scholar
  2. 2.
    Albers, S., Lenzner, P.: On Approximate Nash Equilibria in Network Design. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 14–25. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Alon, N., Demaine, E.D., Hajiaghayi, M., Leighton, T.: Basic network creation games. In: Proceedings of the 22nd ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2010, pp. 106–113. ACM, New York (2010)Google Scholar
  4. 4.
    Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for k-median and facility location problems. SIAM J. Comput. 33(3), 544–562 (2004)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cord-Landwehr, A., Hüllmann, M., Kling, P., Setzer, A.: Basic Network Creation Games with Communication Interests. In: Serna, M. (ed.) SAGT 2012. LNCS, vol. 7615, pp. 72–83. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    Demaine, E.D., Hajiaghayi, M., Mahini, H., Zadimoghaddam, M.: The price of anarchy in cooperative network creation games. SIGecom Exch. 8(2), 2:1–2:20 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Demaine, E.D., Hajiaghayi, M.T., Mahini, H., Zadimoghaddam, M.: The price of anarchy in network creation games. ACM Trans. on Algorithms 8(2), 13 (2012)MathSciNetGoogle Scholar
  8. 8.
    Ehsani, S., Fazli, M., Mehrabian, A., Sadeghian Sadeghabad, S., Safari, M., Saghafian, M., ShokatFadaee, S.: On a bounded budget network creation game. In: Proceedings of the 23rd ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2011, pp. 207–214. ACM, New York (2011)Google Scholar
  9. 9.
    Fabrikant, A., Luthra, A., Maneva, E., Papadimitriou, C.H., Shenker, S.: On a network creation game. In: Proc. of the 22nd Annual Symp. on Principles of Distributed Computing, PODC 2003, pp. 347–351. ACM, New York (2003)Google Scholar
  10. 10.
    Gulyás, A., Kõrösi, A., Szabó, D., Biczók, G.: On greedy network formation. In: Proceedings of ACM SIGMETRICS Performance W-PIN (2012)Google Scholar
  11. 11.
    Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems. ii: The p-medians. SIAM J. on Appl. Math. 37(3), 539–560 (1979)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Lenzner, P.: On Dynamics in Basic Network Creation Games. In: Persiano, G. (ed.) SAGT 2011. LNCS, vol. 6982, pp. 254–265. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Mihalák, M., Schlegel, J.C.: The Price of Anarchy in Network Creation Games Is (Mostly) Constant. In: Kontogiannis, S., Koutsoupias, E., Spirakis, P.G. (eds.) SAGT 2010. LNCS, vol. 6386, pp. 276–287. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Mihalák, M., Schlegel, J.: Asymmetric Swap-Equilibrium: A Unifying Equilibrium Concept for Network Creation Games. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 693–704. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Nash, J.F.: Equilibrium points in n-person games. PNAS 36(1), 48–49 (1950)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Vazirani, V.V.: Approximation algorithms. Springer (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Pascal Lenzner
    • 1
  1. 1.Department of Computer ScienceHumboldt-Universität zu BerlinGermany

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