Skip to main content
SpringerLink
Log in

A Unified Framework for Designing EPTAS for Load Balancing on Parallel Machines

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We consider a general load balancing problem on parallel machines. Our machine environment in particular generalizes the standard models of identical machines, and the model of uniformly related machines, as well as machines with a constant number of types, and machines with activation costs. The objective functions that we consider contain in particular the makespan objective and the minimization of the \(\ell _p\)-norm of the vector of loads of the machines, and each case allows the possibility of job rejection. We consider this general model and design an efficient polynomial time approximation scheme (EPTAS) that applies for all its previously-studied special cases. This EPTAS improves the current best approximation scheme for some of these cases where only a polynomial time approximation scheme was known into an EPTAS.

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. Alon, N., Azar, Y., Woeginger, G.J., Yadid, T.: Approximation schemes for scheduling on parallel machines. J. Sched. 1(1), 55–66 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bansal, N., Pruhs, K.R.: Server scheduling to balance priorities, fairness, and average quality of service. SIAM J. Comput. 39(7), 3311–3335 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bartal, Y., Leonardi, S., Marchetti-Spaccamela, A., Sgall, J., Stougie, L.: Multiprocessor scheduling with rejection. SIAM J. Discrete Math. 13(1), 64–78 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bonifaci, V., Wiese, A.: Scheduling unrelated machines of few different types (2012). arXiv:1205.0974

  5. Cesati, M., Trevisan, L.: On the efficiency of polynomial time approximation schemes. Inf. Process. Lett. 64(4), 165–171 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, L., Jansen, K., Zhang, G.: On the optimality of exact and approximation algorithms for scheduling problems. J. Comput. Syst. Sci. 96, 1–32 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  8. Epstein, L., Levin, A.: An efficient polynomial time approximation scheme for load balancing on uniformly related machines. Math. Program. 147, 1–23 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Epstein, L., Levin, A.: Minimum total weighted completion time: faster approximation schemes (2014). arXiv:1404.1059

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)

    MATH  Google Scholar 

  12. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)

    MATH  Google Scholar 

  13. Gehrke, J.C., Jansen, K., Kraft, S.E.J., Schikowski, J.: A PTAS for scheduling unrelated machines of few different types. In: Proceeding of the 42nd International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM’16), pp. 290–301 (2016)

  14. Hochbaum, D.S.: Various notions of approximations: good, better, best and more. In: Hochbaum, D.S. (ed.) Approximation Algorithms. PWS Publishing Company, Boston (1997)

    Google Scholar 

  15. 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 

  16. 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  MATH  Google Scholar 

  17. Jansen, K.: An EPTAS for scheduling jobs on uniform processors: using an MILP relaxation with a constant number of integral variables. SIAM J. Discrete Math. 24(2), 457–485 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jansen, K., Klein, K.-M., Verschae, J.: Closing the gap for makespan scheduling via sparsification techniques. In: Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP’16), vol. 72, pp. 1–13 (2016)

  19. Jansen, K., Maack, M.: An EPTAS for scheduling on unrelated machines of few different types. In: Proceedings of the 15th International Symposium on Algorithms and Data Structures (WADS’17), pp. 497–508 (2017)

  20. Jansen, K., Maack, M.: An EPTAS for scheduling on unrelated machines of few different types. CoRR (2017). arXiv:1701.03263 (v2)

  21. Jansen, K., Maack, M., Rau, M.: Approximation schemes for machine scheduling with resource (in-)dependent processing times. In: Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’16), pp. 1526–1542 (2016)

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

  23. Khuller, S., Li, J., Saha, B.: Energy efficient scheduling via partial shutdown. In: Proceedings of 21st Symposium on Discrete Algorithms (SODA), pp. 1360–1372. ACM/SIAM (2010)

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Marx, D.: Parameterized complexity and approximation algorithms. Comput. J. 51(1), 60–78 (2008)

    Article  Google Scholar 

  26. Schuurman, P., Woeginger, G.J.: Approximation schemes—a tutorial (2001). http://www.win.tue.nl/~gwoegi/papers/ptas.pdf

  27. Wiese, A., Bonifaci, V., Baruah, S.K.: Partitioned EDF scheduling on a few types of unrelated multiprocessors. Real-Time Syst. 49(2), 219–238 (2013)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Ishai Kones & Asaf Levin

Authors
  1. Ishai Kones
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Asaf Levin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Asaf Levin.

Additional information

Publisher's Note

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

This research was supported by a grant from the GIF, the German-Israeli Foundation for Scientific Research and Development (Grant Number I-1366-407.6/2016). An earlier extended abstract version of this work appears in Proc. CiE 2018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kones, I., Levin, A. A Unified Framework for Designing EPTAS for Load Balancing on Parallel Machines. Algorithmica 81, 3025–3046 (2019). https://doi.org/10.1007/s00453-019-00566-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-019-00566-9

Keywords

Search

Navigation