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Fair Cost-Sharing Methods for Scheduling Jobs on Parallel Machines

  • Yvonne Bleischwitz
  • Burkhard Monien
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3998)

Abstract

We consider the problem of sharing the cost of scheduling n jobs on m parallel machines among a set of agents. In our setting, each agent owns one job and the cost is given by the makespan of the computed assignment. We focus on α-budget-balanced cross-monotonic cost-sharing methods since they guarantee the two substantial mechanism properties α-budget-balance and group-strategyproofness and provide fair cost-shares. For identical jobs on related machines and for arbitrary jobs on identical machines, we give (m+1)/(2m)-budget-balanced cross-monotonic cost-sharing methods and show that this is the best approximation possible. As our major result, we prove that the approximation factor for cross-monotonic cost-sharing methods is unbounded for arbitrary jobs and related machines. We therefore develop a cost-sharing method in the (m+1)/(2m)-core, a weaker but also fair solution concept. We close with a strategyproof mechanism for the model of arbitrary jobs and related machines that recovers at least 3/5 of the cost. All given solutions can be computed in polynomial time.

Keywords

Nash Equilibrium Schedule Problem Completion Time Parallel Machine Steiner Tree 
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.

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References

  1. 1.
    Archer, A., Feigenbaum, J., Krishnamurthy, A., Sami, R.: Approximation and collusion in multicast cost sharing. Games and Economic Behaviour 47, 36–71 (2004)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Archer, A., Tardos, E.: Truthful mechanisms for one-parameter agents. In: Proceedings of the 42th IEEE Symposium on Foundations of Computer Science, pp. 482–491 (2001)Google Scholar
  3. 3.
    Beccetti, L., Könemann, J., Leonardi, S., Pál, M.: Sharing the cost more efficiently: improved approximation for multicommodity rent-or-buy. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 375–384 (2005)Google Scholar
  4. 4.
    Czumaj, A.: Selfish Routing on the Internet. Handbook of Scheduling: Algorithms, Models, and Performance Analysis, ch. 42 (2004)Google Scholar
  5. 5.
    Devanur, N., Mihail, M., Vazirani, V.: Strategyproof cost sharing mechanisms for set cover and facility location problems. In: Proceedings of ACM Conference on Electronic Commerce, pp. 108–114 (2003)Google Scholar
  6. 6.
    Feigenbaum, J., Krishnamurthy, A., Sami, R., Shenker, S.: Hardness results for multicast cost sharing. Theoretical Computer Science 304(1-3), 215–236 (2003)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Feigenbaum, J., Papadimitriou, C., Shenker, S.: Sharing the cost of multicast transmissions. Journal of Computer and System Sciences 63, 21–41 (2001)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Friesen, D.: Tighter bounds for the multifit processor scheduling algorithm. SIAM Journal on Computing 13(1), 170–181 (1984)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Friesen, D.: Tighter bounds for lpt scheduling on uniform processors. SIAM Journal on Computing 16(3), 554–560 (1987)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Friesen, D., Langston, M.: Bounds for multifit scheduling on uniform processors. SIAM Journal on Computing 12(1), 60–70 (1983)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gairing, M., Lücking, T., Monien, B., Tiemann, K.: Nash Equilibria, the Price of Anarchy and the Fully Mixed Nash Equilibrium Conjecture. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 51–65. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Graham, R.: Bounds on multiprocessing timing anomalies. SIAM Journal of Applied Mathematics 17(2), 416–429 (1969)MATHCrossRefGoogle Scholar
  13. 13.
    Gupta, A., Srinivasan, A., Tardos, E.: Cost-Sharing Mechanisms for Network Design. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 139–150. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Hochbaum, D., Shmoys, D.: Using dual approximation algorithms for scheduling problems: theoretical and practical results. Journal of the ACM 34(1), 144–162 (1987)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Hochbaum, D., Shmoys, D.: A polynomial approximation scheme for scheuduling on uniform processors: using the dual approximation approach. SIAM Journal on Computing 17(3), 539–551 (1988)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Horowitz, E., Sahni, S.: Exact and approximate algorithms for scheduling nonidentical processors. Journal of the Association for Computing Machinery 23(2), 317–327 (1976)MATHMathSciNetGoogle Scholar
  17. 17.
    Immorlica, N., Mahdian, M., Mirrokni, V.: Limitations of cross-monotonic cost sharing schemes. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 602–611 (2005)Google Scholar
  18. 18.
    Jain, K., Vazirani, V.: Applications of approximate algorithms to cooperative games. In: Proceedings of the 33th Annual ACM Symposium on Theory of Computing, pp. 364–372 (2001)Google Scholar
  19. 19.
    Kent, K., Skorin-Kapov, D.: Population monotonic cost allocation on msts. In: Operational Research Proceedings KOI, pp. 43–48 (1996)Google Scholar
  20. 20.
    Könemann, J., Leonardi, S., Schäfer, G.: A group-strategyproof mechanism for steiner forests. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 612–619 (2005)Google Scholar
  21. 21.
    Könemann, J., Leonardi, S., Schäfer, G., van Zwam, S.: From primal-dual to cost shares and back: a stronger LP relaxation for the steiner forest problem. In: Proceedings of the 32th Int. Colloquium on Automata, Languages, and Programming, pp. 930–942 (2005)Google Scholar
  22. 22.
    Lenstra, J.K., Shmoys, D.B., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. In: Proceedings of the 28th Annual Symposium on Foundations of Computer Science (FOCS 1987), pp. 217–224 (1987)Google Scholar
  23. 23.
    Leonardi, S., Schäfer, G.: Cross-monotonic cost-sharing methods for connected facility location games. In: ACM Conference on Electronic Commerce, pp. 224–243 (2004)Google Scholar
  24. 24.
    Mishra, D., Rangarajan, B.: Cost sharing in a job scheduling problem using the shapley value. In: Proceedings of the 6th ACM Conference on Electronic Commerce, pp. 232–239 (2005)Google Scholar
  25. 25.
    Moulin, H., Shenker, S.: Strategyproof sharing of submodular costs: budget balance versus efficiency. Economic Theory 18, 511–533 (2001)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Nisan, N., Ronen, A.: Algorithmic Mechanism Design. Games and Economic Behaviour 35, 166–196 (2001); Extended abstract appeard at STOC 1999Google Scholar
  27. 27.
    Pál, M., Tardos, E.: Group strategyproof mechanisms via primal-dual algorithms. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 584–593 (2003)Google Scholar
  28. 28.
    Penna, P., Ventre, C.: The Algorithmic Structure of Group Strategyproof Budget-Balanced Cost-Sharing Mechanisms. In: Proceedings of the 23rd International Symposium on Theoretical Aspects of Computer Science (to appear, 2006)Google Scholar
  29. 29.
    Shapley, L.S.: On balanced sets and cores. Naval Research Logistics Quarterly 14, 453–460 (1967)CrossRefGoogle Scholar
  30. 30.
    Shchepin, E.V., Vakhania, N.: An optimal rounding gives a better approximation for scheduling unrelated machines. Operations Research Letters 33, 127–133 (2005)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Yvonne Bleischwitz
    • 1
    • 2
  • Burkhard Monien
    • 1
  1. 1.Faculty of Computer Science, Electrical Engineering and MathematicsUniversity of PaderbornPaderbornGermany
  2. 2.International Graduate School of Dynamic Intelligent Systems 

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