Mathematical Programming

, Volume 146, Issue 1–2, pp 185–218 | Cite as

Smoothed performance guarantees for local search

  • Tobias Brunsch
  • Heiko Röglin
  • Cyriel Rutten
  • Tjark Vredeveld
Full Length Paper Series A

Abstract

We study popular local search and greedy algorithms for standard machine scheduling problems. The performance guarantee of these algorithms is well understood, but the worst-case lower bounds seem somewhat contrived and it is questionable whether they arise in practical applications. To find out how robust these bounds are, we study the algorithms in the framework of smoothed analysis, in which instances are subject to some degree of random noise. While the lower bounds for all scheduling variants with restricted machines are rather robust, we find out that the bounds are fragile for unrestricted machines. In particular, we show that the smoothed performance guarantee of the jump and the lex-jump algorithm are (in contrast to the worst case) independent of the number of machines. They are \(\Theta (\phi )\) and \(\Theta (\log \phi )\), respectively, where \(1/\phi \) is a parameter measuring the magnitude of the perturbation. The latter immediately implies that also the smoothed price of anarchy is \(\Theta (\log \phi )\) for routing games on parallel links. Additionally, we show that for unrestricted machines also the greedy list scheduling algorithm has an approximation guarantee of  \(\Theta (\log \phi )\).

Mathematics Subject Classification

68Q25 68Q87 68W25 68W40 90B35 

References

  1. 1.
    Angel, E.: A survey of approximation results for local search algorithms. In: Bampis, E., Jansen, K., Kenyon, C. (eds.) Efficient Approximation and Online Algorithms, Volume 3484 of LNCS, pp. 30–73. Springer, Heidelberg (2006)Google Scholar
  2. 2.
    Aspnes, J., Azar, Y., Fiat, A., Plotkin, S.A., Waarts, O.: On-line routing of virtual circuits with applications to load balancing and machine scheduling. J. ACM 44(3), 486–504 (1997)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Awerbuch, B., Azar, Y., Richter, Y., Tsur, D.: Tradeoffs in worst-case equilibria. Theor. Comput. Sci. 361, 200–209 (2006)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Becchetti, L., Leonardi, S., Marchetti-Spaccamela, A., Schäfer, G., Vredeveld, T.: Average case and smoothed competitive analysis for the multi-level feedback algorithm. Math. Oper. Res. 31(3), 85–108 (2006)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Beier, R., Vöcking, B.: Random knapsack in expected polynomial time. J. Comput. Syst. Sci. 69(3), 306–329 (2004)CrossRefMATHGoogle Scholar
  6. 6.
    Cho, Y., Sahni, S.: Bounds for list schedules on uniform processors. SIAM J. Comput. 9, 91–103 (1980)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. ACM Trans. Algorithms 3(1) article 4 (2007)Google Scholar
  8. 8.
    Englert, M., Röglin, H., Vöcking, B.: Worst case and probabilistic analysis of the 2-opt algorithm for the TSP. In: Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1295–13004 (2007)Google Scholar
  9. 9.
    Finn, G., Horowitz, E.: A linear time approximation algorithm for multiprocessor scheduling. BIT 19, 312–320 (1979)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractibility: A Guide to the Theory of NP-Completeness. W.H. Freeman & Co., New York (1979)MATHGoogle Scholar
  11. 11.
    Glass, C.A., Kellerer, H.: Parallel machine scheduling with job assignment restrictions. Naval Res. Logist. 54(3), 250–257 (2007)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45, 1563–1581 (1966)CrossRefGoogle Scholar
  13. 13.
    Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discret. Math. 5, 287–326 (1979)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Hochbaum, D.S., Shmoys, D.B.: A polynomial approximation scheme for machine scheduling on uniform processors: using the dual approximation approach. SIAM J. Comput. 17, 539–551 (1988)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Hoefer, M., Souza, A.: Tradeoffs and average-case equilibria in selfish routing. ACM Trans. Comput. Theory 2(1) article 2 (2010)Google Scholar
  16. 16.
    Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58(301), 13–30 (1963)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Leung, J.Y.T., Li, C.L.: Scheduling with processing set restrictions: a survey. Int. J. Prod. Econ. 116, 251–262 (2008)CrossRefGoogle Scholar
  18. 18.
    Li, C.L.: Scheduling unit-length jobs with machine eligibility restrictions. Eur. J. Oper. Res. 174, 1325–1328 (2006)CrossRefMATHGoogle Scholar
  19. 19.
    Michiels, W.P.A.J., Aarts, E.H.L., Korst, J.H.M.: Theoretical Aspects of Local Search. Springer, Heidelberg (2007)Google Scholar
  20. 20.
    Ou, J., Leung, J.Y.-T., Li, C.L.: Scheduling parallel machines with inclusive set restrictions. Naval Res. Logist. 55(4), 328–338 (2008)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Rutten, C., Recalde, D., Schuurman, P., Vredeveld, T.: Performance guarantees of jump neighborhoods on restricted related parallel machines. Oper. Res. Lett. 40, 287–291 (2012)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Schäfer, G., Sivadasan, N.: Topology matters: smoothed competitiveness of metrical task systems. Theor. Comput. Sci. 341(1–3), 3–14 (2005)Google Scholar
  23. 23.
    Schuurman, P., Vredeveld, T.: Performance guarantees of local search for multiprocessor scheduling. Inf. J. Comput. 19(1), 52–63 (2007)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Spielman, D.A., Teng, S.H.: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM 51(3), 385–463 (2004)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Spielman, D.A., Teng, S.H.: Smoothed analysis: an attempt to explain the behavior of algorithms in practice. Commun. ACM 52(10), 76–84 (2009)CrossRefGoogle Scholar
  26. 26.
    Vöcking, B.: Selfish load balancing. In: Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V. (eds.) Algorithmic Game Theory, chapter 20. Cambridge University Press, New York (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  • Tobias Brunsch
    • 1
  • Heiko Röglin
    • 1
  • Cyriel Rutten
    • 2
  • Tjark Vredeveld
    • 2
  1. 1.Department of Computer ScienceUniversity of BonnBonnGermany
  2. 2.Department of Quantitative EconomicsMaastricht UniversityMaastrichtThe Netherlands

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