Bounds for the Convergence Time of Local Search in Scheduling Problems

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10123)

Abstract

We study the convergence time of local search for a standard machine scheduling problem in which jobs are assigned to identical or related machines. Local search corresponds to the best response dynamics that arises when jobs selfishly try to minimize their costs. We assume that each machine runs a coordination mechanism that determines the order of execution of jobs assigned to it. We obtain various new polynomial and pseudo-polynomial bounds for the well-studied coordination mechanisms Makespan and Shortest-Job-First, using worst-case and smoothed analysis. We also introduce a natural coordination mechanism FIFO, which takes into account the order in which jobs arrive at a machine, and study both its impact on the convergence time and its price of anarchy.

References

  1. 1.
    Beier, R., Vöcking, B.: Random knapsack in expected polynomial time. J. Comput. Syst. Sci. 69(3), 306–329 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brucker, P., Hurink, J., Werner, F.: Models and algorithms for planning and scheduling problems improving local search heuristics for some scheduling problems. Part II. Discrete Appl. Math. 72(1), 47–69 (1997). doi:http://dx.doi.org/10.1016/S0166-218X(96)00036-4 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brunsch, T., Röglin, H., Rutten, C., Vredeveld, T.: Smoothed performance guarantees for local search. Math. Program. 146(1–2, Ser. A), 185–218 (2014). doi:10.1007/s10107-013-0683-7 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Christodoulou, G., Koutsoupias, E., Nanavati, A.: Coordination mechanisms. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 345–357. Springer, Heidelberg (2004). doi:10.1007/978-3-540-27836-8_31 CrossRefGoogle Scholar
  5. 5.
    Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. Trans. Algorithms ACM 3(1) (2007)Google Scholar
  6. 6.
    Etscheid, M.: Performance guarantees for scheduling algorithms under perturbed machine speeds. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) ISAAC 2013. LNCS, vol. 8283, pp. 207–217. Springer, Heidelberg (2013). doi:10.1007/978-3-642-45030-3_20 CrossRefGoogle Scholar
  7. 7.
    Even-Dar, E., Kesselman, A., Mansour, Y.: Convergence time to Nash equilibria. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 502–513. Springer, Heidelberg (2003). doi:10.1007/3-540-45061-0_41 CrossRefGoogle Scholar
  8. 8.
    Feldmann, R., Gairing, M., Lücking, T., Monien, B., Rode, M.: Nashification and the coordination ratio for a selfish routing game. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 514–526. Springer, Heidelberg (2003). doi:10.1007/3-540-45061-0_42 CrossRefGoogle Scholar
  9. 9.
    Finn, G., Horowitz, E.: A linear time approximation algorithm for multiprocessor scheduling. BIT 19, 312–320 (1979)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Complexity results for multiprocessor scheduling under resource constraints. SIAM J. Comput. 4, 397–411 (1975)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Goldberg, P.W.: Bounds for the convergence rate of randomized local search in a multiplayer load-balancing game. In: Proceedings of the PODC 2004, pp. 131–140 (2004). doi:10.1145/1011767.1011787
  12. 12.
    Hurkens, C.A.J., Vredeveld, T.: Local search for multiprocessor scheduling: how many moves does it take to a local optimum? Oper. Res. Lett. 31(2), 137–141 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Immorlica, N., Li, L., Mirrokni, V.S., Schulz, A.S.: Coordination mechanisms for selfish scheduling. Theor. Comput. Sci. 410(17), 1589–1598 (2009). doi:10.1016/j.tcs.2008.12.032 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Manthey, B., Röglin, H.: Smoothed analysis: analysis of algorithms beyond worst case. IT - Information Technology 53(6), 280–286 (2011)CrossRefGoogle Scholar
  15. 15.
    Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Internat. J. Game Theor. 2, 65–67 (1973)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Schuurman, P., Vredeveld, T.: Performance guarantees of local search for multiprocessor scheduling. Informs J. Comput. 19(1), 52–63 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Spielman, D., Teng, S.-H.: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM 51(3), 385–463 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Spielman, D., Teng, S.-H.: Smoothed analysis: an attempt to explain the behavior of algorithms in practice. Commun. ACM 52(10), 76–84 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2016

Authors and Affiliations

  • Tobias Brunsch
    • 1
  • Michael Etscheid
    • 1
  • Heiko Röglin
    • 1
  1. 1.Department of Computer ScienceUniversity of BonnBonnGermany

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