Skip to main content
SpringerLink
Log in

Performance guarantees of local search for minsum scheduling problems

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

We study the worst-case performance guarantee of locally optimal solutions for the problem of minimizing the total weighted and unweighted completion time on parallel machine environments. Our method makes use of a mapping that maps a schedule into an inner product space so that the norm of the mapping is closely related to the cost of the schedule. We apply the method to study the most basic local search heuristics for scheduling, namely jump and swap, and establish their worst-case performance in the case of unrelated, restricted related and restricted identical machines.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. When there are no weights, these are taken to be 1, and therefore the rule is just the shortest processing time first rule.

  2. Note that it is possible that \(|{\mathcal {M}}'_j|=1\).

  3. Observe that job j has to be inserted on machine \(x_j^*\) at the appropriate position (defined by WSPT rule).

References

  1. Abed, F., Correa, J.R., Huang, C.: Optimal coordination mechanisms for multi-job scheduling games. In: ESA (2014)

  2. Ahuja, R., Ergun, O., Orlin, J., Punnen, A.: A survey of very large-scale neighborhood search techniques. Discrete Appl. Math. 123(1–3), 75–102 (2002)

    Article  MathSciNet  Google Scholar 

  3. Angel, E.: A survey of approximation results for local search algorithms. In: Bampis, E., Jansen, K., Kenyon, C. (eds.) Efficient Approximation and Online Algorithms: Recent Progress on Classical Combinatorial Optimization Problems and New Applications, vol. 3484, pp. 30–73. Springer, Berlin (2006)

    Chapter  Google Scholar 

  4. Bansal, N., Srinivasan, A., Svensson, O.: Lift-and-round to improve weighted completion time on unrelated machines. In: STOC (2016)

  5. Brucker, P., Hurink, J., Werner, F.: Improving local search heuristics for some scheduling problems. part II. Discrete Appl. Math. 72(1), 47–69 (1997)

    Article  MathSciNet  Google Scholar 

  6. Brueggemann, T., Hurink, J.L., Kern, W.: Quality of move-optimal schedules for minimizing total weighted completion time. Op. Res. Lett. 34(5), 583–590 (2006)

    Article  MathSciNet  Google Scholar 

  7. Brueggemann, T., Hurink, J.L., Vredeveld, T., Woeginger, G.J.: Exponential size neighborhoods for makespan minimization scheduling. Naval Res. Logist. 58(8), 795–803 (2011)

    Article  MathSciNet  Google Scholar 

  8. Bruno, J., Coffman Jr., E.G., Sethi, R.: Scheduling independent tasks to reduce mean finishing time. Commun. ACM 17(7), 382–387 (1974)

    Article  MathSciNet  Google Scholar 

  9. Caragiannis, I., Flammini, M., Kaklamanis, C., Kanellopoulos, P., Moscardelli, L.: Tight bounds for selfish and greedy load balancing. Algorithmica 61(3), 606–637 (2011)

    Article  MathSciNet  Google Scholar 

  10. Cho, Y., Sahni, S.: Bounds for list schedules on uniform processors. SIAM J. Comput. 9(1), 91–103 (1980)

    Article  MathSciNet  Google Scholar 

  11. Cole, R., Correa, J.R., Gkatzelis, V., Mirrokni, V., Olver, N.: Decentralized utilitarian mechanisms for scheduling games. Games Econ. Behav. 92, 306–326 (2015)

    Article  MathSciNet  Google Scholar 

  12. Conway, R., Maxwell, W., Miller, L.: Theory of Scheduling. Addison-Wesley, Reading (1967)

    MATH  Google Scholar 

  13. Correa, J.R., Queyranne, M.: Efficiency of equilibria in restricted uniform machine scheduling with total weighted completion time as social cost. Naval Res. Logist. 59(5), 384–395 (2012)

    Article  MathSciNet  Google Scholar 

  14. Epstein, L., Sgall, J.: Approximation schemes for scheduling on uniformly related and identical parallel machines. Algorithmica 39(1), 43–57 (2004)

    Article  MathSciNet  Google Scholar 

  15. Finn, G., Horowitz, E.: A linear time approximation algorithm for multiprocessor scheduling. BIT Numer. Math. 19(3), 312–320 (1979)

    Article  MathSciNet  Google Scholar 

  16. Frangioni, A., Necciari, E., Scutellà, M.G.: A multi-exchange neighborhood for minimum makespan parallel machine scheduling problems. J. Comb. Optim. 8(2), 195–220 (2004)

    Article  MathSciNet  Google Scholar 

  17. Garey, M.R., Johnson, D.S.: Strong NP-Completeness results: motivation, examples, and implications. J. ACM 25(3), 499–508 (1978)

    Article  MathSciNet  Google Scholar 

  18. Garey, M.R., Johnson, D.S.: Computers and Intractability, a Guide to the Theory of NP-Completness. WH Freeman and Co, San Francisco (1979)

    MATH  Google Scholar 

  19. 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. Discrete Math. 5, 287–326 (1979)

    Article  MathSciNet  Google Scholar 

  20. Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems theoretical and practical results. J. ACM 34(1), 144–162 (1987)

    Article  MathSciNet  Google Scholar 

  21. Hochbaum, D.S., Shmoys, D.B.: A polynomial approximation scheme for scheduling on uniform processors: using the dual approximation approach. SIAM J. Comput. 17(3), 539–551 (1988)

    Article  MathSciNet  Google Scholar 

  22. Horn, W.A.: Technical note-minimizing average flow time with parallel machines. Op. Res. 21(3), 846–847 (1973)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. Comput. Sci. Rev. 3(2), 65–69 (2009)

    Article  Google Scholar 

  25. Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B.: Sequencing and scheduling: algorithms and complexity. Handb. Op. Res. Manag. Sci. 4, 445–522 (1993)

    Google Scholar 

  26. Lenstra, J., Rinnooy Kan, A.H.G., Brucker, P.: Complexity of machine scheduling problems. Ann. Discrete Math. 1, 343–362 (1977)

    Article  MathSciNet  Google Scholar 

  27. Lenstra, J.K., Shmoys, D.B., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46(1–3), 259–271 (1990)

    Article  MathSciNet  Google Scholar 

  28. Li, S.: Scheduling to minimize total weighted completion time via time-indexed linear programming relaxations. In: FOCS (2017)

  29. Michiels, W., Aarts, E., Korst, J.: Theoretical Aspects of Local Search. Springer Science & Business Media, Berlin (2007)

    MATH  Google Scholar 

  30. Pinedo, M.L.: Scheduling: Theory, Algorithms, and Systems. Springer, Berlin (2016)

    Book  Google Scholar 

  31. Recalde, D., Rutten, C., Schuurman, P., Vredeveld, T.: Local search performance guarantees for restricted related parallel machine scheduling. In: LATIN (2010)

  32. Rutten, C., Recalde, D., Schuurman, P., Vredeveld, T.: Performance guarantees of jump neighborhoods on restricted related parallel machines. Op. Res. Lett. 40(4), 287–291 (2012)

    Article  MathSciNet  Google Scholar 

  33. Schuurman, P., Vredeveld, T.: Performance guarantees of local search for multiprocessor scheduling. INFORMS J. Comput. 19(1), 52–63 (2007)

    Article  MathSciNet  Google Scholar 

  34. Skutella, M., Woeginger, G.J.: A ptas for minimizing the total weighted completion time on identical parallel machines. Math. Op. Res. 25(1), 63–75 (2000)

    Article  MathSciNet  Google Scholar 

  35. Smith, W.E.: Various optimizers for single-stage production. Naval Res. Logist. 3(1–2), 59–66 (1956)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank Fidaa Abed for providing the instance presented in § 5.4. We also thank two anonymous referees for many helpful suggestions that greatly improved the presentation of the paper. This work was partially supported by ANID Chile through grants BASAL AFB-180003 and BASAL AFB-170001.

Author information

Authors and Affiliations

  1. Department of Industrial Engineering, Universidad de Chile, Santiago, Chile

    José R. Correa

  2. Department of Industrial Engineering, Universidad del Bío-Bío, Concepción, Chile

    Felipe T. Muñoz

Authors
  1. José R. Correa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Felipe T. Muñoz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José R. Correa.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Correa, J.R., Muñoz, F.T. Performance guarantees of local search for minsum scheduling problems. Math. Program. 191, 847–869 (2022). https://doi.org/10.1007/s10107-020-01571-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-020-01571-5

Keywords

Mathematics Subject Classification

Search

Navigation