The Price of Anarchy for Minsum Related Machine Scheduling

  • Ruben Hoeksma
  • Marc Uetz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7164)


We address the classical uniformly related machine scheduling problem with minsum objective. The problem is solvable in polynomial time by the algorithm of Horowitz and Sahni. In that solution, each machine sequences its jobs shortest first. However when jobs may choose the machine on which they are processed, while keeping the same sequencing rule per machine, the resulting Nash equilibria are in general not optimal. The price of anarchy measures this optimality gap. By means of a new characterization of the optimal solution, we show that the price of anarchy in this setting is bounded from above by 2. We also give a lower bound of e/(e − 1) ≈ 1.58. This complements recent results on the price of anarchy for the more general unrelated machine scheduling problem, where the price of anarchy equals 4. Interestingly, as Nash equilibria coincide with shortest processing time first (SPT) schedules, the same bounds hold for SPT schedules. Thereby, our work also fills a gap in the literature.


Nash Equilibrium Completion Time Optimal Schedule Coordination Mechanism Short Processing Time 
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  1. 1.
    Aumann, R.J.: Subjectivity and correlation in randomized strategies. J. Math. Econom. 1(1), 67–96 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Azar, Y., Jain, K., Mirrokni, V.: (Almost) optimal coordination mechanisms for unrelated machine scheduling. In: Proceedings 19th SODA, pp. 323–332. ACM/SIAM (2008)Google Scholar
  3. 3.
    Christodoulou, G., Koutsoupias, E., Nanavati, A.: Coordination mechanisms. Theoret. Comput. Sci. 410(36), 3327–3336 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cole, R., Correa, J.R., Gkatzelis, V., Mirrokni, V., Olver, N.: Inner Product Spaces for MinSum Coordination Mechanisms. In: Proceedings 43rd STOC, pp. 539–548. ACM (2011)Google Scholar
  5. 5.
    Conway, R.W., Maxwell, W.L., Miller, L.W.: Theory of Scheduling. Addison-Wesley Publishing Co., Reading (1967)zbMATHGoogle Scholar
  6. 6.
    Correa, J., Queyranne, M.: Efficiency of Equilibria in Restricted Uniform Machine Scheduling with MINSUM Social Cost (manuscript) (2010)Google Scholar
  7. 7.
    Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. ACM Trans. Algorithms 3(1), Art. 4, 17 (2007)Google Scholar
  8. 8.
    Graham, R., Lawler, E., Lenstra, J., Rinnooy Kan, A.: Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics 5(2), 287–326 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Heydenreich, B., Müller, R., Uetz, M.: Games and mechanism design in machine scheduling - An introduction. Production and Operations Management 16(4), 437–454 (2007)CrossRefGoogle Scholar
  10. 10.
    Horowitz, E., Sahni, S.: Exact and approximate algorithms for scheduling nonidentical processors. Journal of the ACM 23(2), 317–327 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Ibarra, O., Kim, C.: Heuristic algorithms for scheduling independent tasks on nonidentical processors. Journal of the ACM 24(2), 280–289 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Immorlica, N., Li, L., Mirrokni, V.S., Schulz, A.S.: Coordination mechanisms for selfish scheduling. Theoret. Comput. Sci. 410(17), 1589–1598 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Koutsoupias, E., Papadimitriou, C.: Worst-Case Equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  14. 14.
    Myerson, R.B.: Utilitarianism, egalitarianism, and the timing effect in social choice problems. Econometrica 49(4), 883–897 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Myerson, R.B.: Game theory - Analysis of conflict. Harvard University Press, Cambridge (1991)zbMATHGoogle Scholar
  16. 16.
    Papadimitriou, C.: Algorithms, games, and the internet. In: Proceedings 33rd STOC, pp. 749–753. ACM (2001)Google Scholar
  17. 17.
    Roughgarden, T.: Intrinsic robustness of the price of anarchy. In: Proceedings 41st STOC, pp. 513–522. ACM (2009)Google Scholar
  18. 18.
    Yu, L., She, K., Gong, H., Yu, C.: Price of anarchy in parallel processing. Inform. Process. Lett. 110(8-9), 288–293 (2010)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ruben Hoeksma
    • 1
  • Marc Uetz
    • 1
  1. 1.Dept. Applied MathematicsUniversity of TwenteEnschedeThe Netherlands

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