Skip to main content
SpringerLink
Log in

Online Minimization of the Maximum Starting Time: Migration Helps

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We consider non-preemptive load balancing on m identical machines where the cost of a machine is defined as the maximum starting time of a job assigned to the machine, and the goal is to find a partition of the jobs that minimizes the maximum machine cost. In our variant the last job on each machine is the smallest job assigned to that machine. The online model for this problem is too restrictive as a trivial example shows that there is no competitive algorithm for the problem. We show that a constant migration factor is sufficient to guarantee a \((\frac{3}{2} +\varepsilon )\)-competitive algorithm for all \(\varepsilon >0\), and using a constant migration factor cannot lead to a better than a \(\frac{3}{2}\)-competitive algorithm. We also show that for this problem, constant amortized migration factor is strictly more powerful and allows us to obtain a polynomial time approximation scheme with a constant amortized migration factor. Thus, the ability to move some limited set of jobs on each step allows the algorithm to be much better than in the pure online settings.

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

Similar content being viewed by others

References

  1. Berndt, S., Dreismann, V., Grage, K., Jansen, K., Knof, I.: Robust online algorithms for certain dynamic packing problems. In Proceedings of the 17th International Workshop on Approximation and Online Algorithms, (WAOA 2019), pp. 43–59 (2019)

  2. Berndt, S., Epstein, L., Jansen, K., Levin, A., Maack, M., Rohwedder, L.: Online bin covering with limited migration. In: Proceedings of the 27th Annual European Symposium on Algorithms, ESA 2019, vol. 144, pp. 18:1–18:14 (2019)

  3. Berndt, S., Jansen, K., Klein, K.: Fully dynamic bin packing revisited. Math. Program. 179(1), 109–155 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cook, W., Gerards, A., Schrijver, A., Tardos, É.: Sensitivity theorems in integer linear programming. Math. Program. 34(3), 251–264 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. Epstein, L., Levin, A.: Asymptotic fully polynomial approximation schemes for variants of open-end bin packing. Inf. Process. Lett. 109(1), 32–37 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Epstein, L., Levin, A.: A robust APTAS for the classical bin packing problem. Math. Program. 119(1), 33–49 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Epstein, L., Levin, A.: Robust approximation schemes for cube packing. SIAM J. Optim. 23(2), 1310–1343 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Epstein, L., Levin, A.: Robust algorithms for preemptive scheduling. Algorithmica 69(1), 26–57 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Epstein, L., Levin, A.: Robust algorithms for total completion time. Discrete Optim. 33, 70–86 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Epstein, L., Levin, A.: Starting time minimization for the maximum job variant. Discrete Appl. Math. 307, 79–87 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  11. Epstein, L., Tassa, T.: Approximation schemes for the min–max starting time problem. Acta Inform. 40(9), 657–674 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Epstein, L., van Stee, R.: Minimizing the maximum starting time on-line. Inf. Comput. 195(1–2), 53–65 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Feldkord, B., Feldotto, M., Gupta, A., Guruganesh, G., Kumar, A., Riechers, S., Wajc, D.: Fully-dynamic bin packing with little repacking. In: Proceedings of the 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, pp. 51:1–51:24 (2018)

  14. Gálvez, W., Soto, J.A., Verschae, J.: Symmetry exploitation for online machine covering with bounded migration. ACM Trans. Algorithms 16(4), 43:1-43:22 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jansen, K., Klein, K.: A robust AFPTAS for online bin packing with polynomial migration. SIAM J. Discrete Math. 33(4), 2062–2091 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jansen, K., Klein, K., Kosche, M., Ladewig, L.: Online strip packing with polynomial migration. In: Proceedings of the 20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017, pp. 13:1–13:18 (2017)

  17. Kannan, R.: Improved algorithms for integer programming and related lattice problems. In: Proceedings of the 15th Annual ACM Symposium on Theory of Computing, STOC 1983, pp. 193–206 (1983)

  18. Lenstra, H.W., Jr.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Leung, J.Y.-T., Dror, M., Young, G.H.: A note on an open-end bin packing problem. J. Sched. 4(4), 201–207 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lu, L., Zhang, L.: Online single machine scheduling to minimize the maximum starting time. Asia-Pac. J. Oper. Res. 34(5). Paper number 1750022, 9 pp (2017)

  21. Sanders, P., Sivadasan, N., Skutella, M.: Online scheduling with bounded migration. Math. Oper. Res. 34(2), 481–498 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Hoboken (1986)

    MATH  Google Scholar 

  23. Skutella, M., Verschae, J.: Robust polynomial-time approximation schemes for parallel machine scheduling with job arrivals and departures. Math. Oper. Res. 41(3), 991–1021 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Verschae, J.: Robust scheduling algorithms. In: Encyclopedia of Algorithms, pp. 1863–1865 (2016)

  25. Zhang, G.: Parameterized on-line open-end bin packing. Computing 60(3), 267–274 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

Partially supported by a grant from GIF - the German-Israeli Foundation for Scientific Research and Development (Grant Number I-1366-407.6/2016) and by grants from ISF - Israeli Science Foundation (Grant Numbers 308/18 and 1467/22).

Author information

Authors and Affiliations

  1. Faculty of Industrial Engineering and Management, The Technion, 32000, Haifa, Israel

    Asaf Levin

Authors
  1. Asaf Levin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Asaf Levin.

Ethics declarations

Conflict of interest

The author has no relevant financial or non-financial interests to disclose. The author has no competing interests to declare that are relevant to the content of this article.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Levin, A. Online Minimization of the Maximum Starting Time: Migration Helps. Algorithmica 85, 2238–2259 (2023). https://doi.org/10.1007/s00453-023-01097-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-023-01097-0

Keywords

Search

Navigation