Advertisement

Non-cooperative Tree Creation

Extended Abstract
  • Martin Hoefer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)

Abstract

In this paper we consider the connection game, a simple network design game with independent selfish agents that was introduced by Anshelevich et al [4]. In addition we present a generalization called backbone game to model hierarchical network and backbone link creation between existing network structures. In contrast to the connection game each player considers a number of groups of terminals and wants to connect at least one terminal from each group into a network. In both games we focus on an important subclass of tree games, in which every feasible network is guaranteed to be connected.

For tree connection games, in which every player holds 2 terminals, we show that there is a Nash equilibrium as cheap as the optimum network. We give a polynomial time algorithm to find a cheap (2+ε)-approximate Nash equilibrium, which can be generalized to a cheap (3.1+ε)-approximate Nash equilibrium for the case of any number of terminals per player. This improves the guarantee of the only previous algorithm for the problem [4], which returns a (4.65+ε)-approximate Nash equilibrium. Tightness results for the analysis of all algorithms are derived.

For single source backbone games, in which each player wants to connect one group to a common source, there is a Nash equilibrium as cheap as the optimum network and a polynomial time algorithm to find a cheap (1+ε)-approximate Nash equilibrium.

Keywords

Nash Equilibrium Polynomial Time Algorithm Steiner Tree Optimum Network Steiner Tree Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agrawal, A., Klein, P., Ravi, R.: When trees collide: An approximation algorithm for the generalized Steiner problem on networks. SIAM J Comp 24(3), 445–456 (1995)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Albers, S., Eilts, S., Even-Dar, E., Mansour, Y., Roditty, L.: On nash equilibria for a network creation game. In: Proc 17th Ann ACM-SIAM Symp Discrete Algorithms (SODA), pp. 89–98 (2006)Google Scholar
  3. 3.
    Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, É., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. In: Proc 45th Ann IEEE Symp Foundations Comp Sci (FOCS), pp. 295–304 (2004)Google Scholar
  4. 4.
    Anshelevich, E., Dasgupta, A., Tardos, É., Wexler, T.: Near-optimal network design with selfish agents. In: Proc 35th Ann ACM Symp Theo Comp (STOC), pp. 511–520 (2003)Google Scholar
  5. 5.
    Corbo, J., Parkes, D.: The price of selfish behavior in bilateral network formation. In: Proc 24th Ann ACM Symp Principles of Distributed Comp, PODC (2005)Google Scholar
  6. 6.
    Czumaj, A., Krysta, P., Vöcking, B.: Selfish traffic allocation for server farms. In: Proc 34th Ann ACM Symp Theory Comp (STOC), pp. 287–296 (2002)Google Scholar
  7. 7.
    Fabrikant, A., Luthera, A., Maneva, E., Papadimitriou, C., Shenker, S.: On a network creation game. In: Proc 22nd Ann ACM Symp Principles of Distributed Comp (PODC), pp. 347–351 (2003)Google Scholar
  8. 8.
    Garg, N., Konjevod, G., Ravi, R.: A polylogarithmic approximation algorithm for the Group Steiner tree problem. J Algorithms 37, 66–84 (2000)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Goemams, M., Williamson, D.: A general approximation technique for constrained forest problems. SIAM J Comp 24(2), 296–317 (1995)CrossRefGoogle Scholar
  10. 10.
    Hoefer, M., Krysta, P.: Geometric network design with selfish agents. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 167–178. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Jackson, M.: A survey of models of network formation: Stability and efficiency. In: Demange, G., Wooders, M. (eds.) Group Formation in Economics; Networks, Clubs and Coalitions, ch. 1. Cambridge University Press, Cambridge (2004)Google Scholar
  12. 12.
    Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: Proc 16th Ann Symp Theoretical Aspects Comp Sci (STACS), pp. 404–413 (1999)Google Scholar
  13. 13.
    Reich, G., Widmayer, P.: Beyond Steiner’s problem: A VLSI oriented generalization. In: Nagl, M. (ed.) WG 1989. LNCS, vol. 411, pp. 196–210. Springer, Heidelberg (1990)Google Scholar
  14. 14.
    Robins, G., Zelikovsky, A.: Improved Steiner tree approximation in graphs. In: Proc 10th Ann ACM-SIAM Symp Discrete Algorithms (SODA), pp. 770–779 (2000)Google Scholar
  15. 15.
    Roughgarden, T., Tardos, É.: How bad is selfish routing? J ACM 49(2), 236–259 (2002)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Schulz, A., Stier Moses, N.: Selfish routing in capacitated networks. Math Oper Res 29(4), 961–976 (2004)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Vetta, A.: Nash equilibria in competitive societies with application to facility location, traffic routing and auctions. In: Proc 43rd Ann IEEE Symp Foundations Comp Sci (FOCS), p. 416 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Martin Hoefer
    • 1
  1. 1.Department of Computer & Information ScienceKonstanz UniversityKonstanzGermany

Personalised recommendations