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Computing Nash Equilibria for Scheduling on Restricted Parallel Links

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Abstract

We consider the problem of routing n users on m parallel links under the restriction that each user may only be routed on a link from a certain set of allowed links for the user. So, this problem is equivalent to the correspondingly restricted scheduling problem of assigning n jobs to m parallel machines. In a Nash equilibrium, no user may improve its own Individual Cost (latency) by unilaterally switching to another link from its set of allowed links.

For identical links, we present, as our main result, a polynomial time algorithm to compute from any given assignment a Nash equilibrium with non-increased makespan. The algorithm gradually transforms the assignment by pushing the unsplittable user traffics through a flow network, which is constructed from the users and the links. The algorithm uses ideas from blocking flows.

Furthermore, we use techniques simular to those in the generic PreflowPush algorithm to approximate in polynomial time a schedule with optimum makespan. This results to an improved approximation factor of \(2-\frac{1}{w_{1}}\) for identical links, where w 1 is the largest user traffic, and to an approximation factor of 2 for related links.

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References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice Hall, New York (1993)

    Google Scholar 

  2. Awerbuch, B., Azar, Y., Richter, Y., Tsur, D.: Tradeoffs in worst-case equilibria. Theor. Comput. Sci. 361(2–3), 200–209 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, X., Deng, X.: Settling the complexity of two-player Nash equilibrium. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 261–272 (2006)

  4. Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 71–78 (2006)

  5. Dinits, E.A.: Algorithm for solution of a problem of maximum flow in a network with power estimation. Sov. Math. Dokl. 11, 1277–1280 (1970)

    MATH  Google Scholar 

  6. Dinitz, Y., Garg, N., Goemans, M.X.: On the single-source unsplittable flow problem. Combinatorica 19(1), 17–41 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Feldmann, R., Gairing, M., Lücking, T., Monien, B., Rode, M.: Nashification and the coordination ratio for a selfish routing game. In: Proceedings of the 30th International Colloquium on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 2719, pp. 514–526. Springer, Berlin (2003)

    Chapter  Google Scholar 

  8. Fotakis, D., Kontogiannis, S., Koutsoupias, E., Mavronicolas, M., Spirakis, P.: The structure and complexity of Nash equilibria for a selfish routing game. In: Proceedings of the 29th International Colloquium on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 2380, pp. 123–134. Springer, Berlin (2002)

    Chapter  Google Scholar 

  9. Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Spirakis, P.: Structure and complexity of extreme Nash equilibria. Theor. Comput. Sci. 343(1–2), 133–157 (2005)

    Article  MATH  Google Scholar 

  10. Gairing, M., Lücking, T., Mavronicolas, M., Monien, B.: The price of anarchy for restricted parallel links. Parallel Process. Lett. 16(1), 117–131 (2006)

    Article  MathSciNet  Google Scholar 

  11. Gairing, M., Monien, B., Woclaw, A.: A faster combinatorial approximation algorithm for scheduling unrelated parallel machines. Theor. Comput. Sci. 380(1–2), 87–99 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Goldberg, A.V.: Efficient graph algorithms for sequential and parallel computers. Ph.D. Thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (1987)

  13. Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum flow problem. J. ACM 35(4), 921–940 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  14. Horowitz, E., Sahni, S.: Exact and approximate algorithms for scheduling nonidentical processors. J. ACM 23(2), 317–327 (1976)

    MATH  MathSciNet  Google Scholar 

  15. Kleinberg, J.: Single-source unsplittable flow. In: Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, pp. 68–77 (1996)

  16. Kolliopoulos, S.G., Stein, C.: Approximation algorithms for single-source unsplittable flow. SIAM J. Comput. 31(3), 919–946 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: Proceedings of the 16th International Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 1563, pp. 404–413. Springer, Berlin (1999)

    Google Scholar 

  18. Lenstra, J.K., Shmoys, D.B., Tardos, É.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46(3), 259–271 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  19. McKelvey, R.D., McLennan, A.: Computation of equilibria in finite games. In: Amman, H., Kendrick, D., Rust, J. (eds.) Handbook of Computational Economics, vol. 1, pp. 87–142 (1996). Chap. 2

  20. Milchtaich, I.: Congestion games with player-specific payoff functions. Games Econ. Behav. 13(1), 111–124 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Nash, J.F.: Equilibrium points in n-person games. In: Proceedings of the National Academy of Sciences of the United States of America, vol. 36, pp. 48–49 (1950)

  22. Nash, J.F.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)

    Article  MathSciNet  Google Scholar 

  23. Papadimitriou, C.H.: Algorithms, games and the Internet. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 749–753 (2001)

  24. Shchepin, E.V., Vakhania, N.: An optimal rounding gives a better approximation for scheduling unrelated machines. Oper. Res. Lett. 33(2), 127–133 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. International Computer Science Institute (ICSI), 1947 Center Street, Berkeley, CA, 94704, USA

    Martin Gairing

  2. Faculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, Fürstenallee 11, 33102, Paderborn, Germany

    Thomas Lücking & Burkhard Monien

  3. Department of Computer Science, University of Cyprus, P.O. Box 20537, Nicosia, 1678, Cyprus

    Marios Mavronicolas

Authors
  1. Martin Gairing
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  2. Thomas Lücking
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  3. Marios Mavronicolas
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  4. Burkhard Monien
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Correspondence to Martin Gairing.

Additional information

A preliminary version of this work appeared in the Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 613–622, June 2004. This work has been partially supported by the IST Program of the European Union under contract numbers IST-1999-14186 ( \(\mathsf{ALCOM-FT}\) ), IST-2001-33116 ( \(\mathsf{FLAGS}\) ), 001907 ( \(\mathsf{DELIS}\) ) and 015964 ( \(\mathsf{AEOLUS}\) ). Parts of the work were done while the last author was visiting the Department of Computer Science, University of Cyprus. Research for this work was done while the first author was at the University of Paderborn.

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Gairing, M., Lücking, T., Mavronicolas, M. et al. Computing Nash Equilibria for Scheduling on Restricted Parallel Links. Theory Comput Syst 47, 405–432 (2010). https://doi.org/10.1007/s00224-009-9191-9

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  • DOI: https://doi.org/10.1007/s00224-009-9191-9

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