Approximating Weighted Completion Time for Order Scheduling with Setup Times

  • Alexander Mäcker
  • Friedhelm Meyer auf der Heide
  • Simon PukropEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12011)


Consider a scheduling problem in which jobs need to be processed on a single machine. Each job has a weight and is composed of several operations belonging to different families. The machine needs to perform a setup between the processing of operations of different families. A job is completed when its latest operation completes and the goal is to minimize the total weighted completion time of all jobs.

We study this problem from the perspective of approximability and provide constant factor approximations as well as an inapproximability result. Prior to this work, only the NP-hardness of the unweighted case and the polynomial solvability of a certain special case were known.


Order scheduling Multioperation jobs Total completion time Approximation Setup times 


  1. 1.
    Allahverdi, A.: The third comprehensive survey on scheduling problems with setup times/costs. Eur. J. Oper. Res. 246(2), 345–378 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Allahverdi, A., Gupta, J.N., Aldowaisan, T.: A review of scheduling research involving setup considerations. Omega 27(2), 219–239 (1999)CrossRefGoogle Scholar
  3. 3.
    Allahverdi, A., Ng, C.T., Cheng, T.C.E., Kovalyov, M.Y.: A survey of scheduling problems with setup times or costs. Eur. J. Oper. Res. 187(3), 985–1032 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bansal, N., Khot, S.: Optimal long code test with one free bit. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 453–462. IEEE (2009)Google Scholar
  5. 5.
    Chekuri, C., Motwani, R.: Precedence constrained scheduling to minimize sum of weighted completion times on a single machine. Discrete Appl. Math. 98(1–2), 29–38 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Correa, J.R., et al.: Strong LP formulations for scheduling splittable jobs on unrelated machines. Math. Program. 154(1–2), 305–328 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Correa, J.R., Verdugo, V., Verschae, J.: Splitting versus setup trade-offs for scheduling to minimize weighted completion time. Oper. Res. Lett. 44(4), 469–473 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Divakaran, S., Saks, M.E.: Approximation algorithms for problems in scheduling with set-ups. Discrete Appl. Math. 156(5), 719–729 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gerodimos, A.E., Glass, C.A., Potts, C.N., Tautenhahn, T.: Scheduling multi-operation jobs on a single machine. Ann. OR 92, 87–105 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hall, L.A., Schulz, A.S., Shmoys, D.B., Wein, J.: Scheduling to minimize average completion time: off-line and on-line approximation algorithms. Math. Oper. Res. 22(3), 513–544 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Happe, M., Meyer auf der Heide, F., Kling, P., Platzner, M., Plessl, C.: On-the-fly computing: a novel paradigm for individualized IT services. In: Proceedings of the 16th IEEE International Symposium on Object/Component/Service-Oriented Real-Time Distributed Computing (ISORC), pp. 1–10. IEEE Computer Society (2013)Google Scholar
  12. 12.
    Jansen, K., Klein, K., Maack, M., Rau, M.: Empowering the configuration-IP-new PTAS results for scheduling with setups times. In: Proceedings of the 10th Innovations in Theoretical Computer Science Conference (ITCS). LIPIcs, vol. 124, pp. 1–19. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2019)Google Scholar
  13. 13.
    Jansen, K., Maack, M., Mäcker, A.: Scheduling on (un-)related machines with setup times. In: Proceedings of the 2019 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pp. 145–154. IEEE Computer Society (2019)Google Scholar
  14. 14.
    Lawler, E.L.: Sequencing jobs to minimize total weighted completion time subject to precedence constraints. Ann. Discrete Math. 2, 75–90 (1978)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lenstra, J.K., Kan, A.H.G.R.: Complexity of scheduling under precedence constraints. Oper. Res. 26(1), 22–35 (1978)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Leung, J.Y., Li, H., Pinedo, M.: Order scheduling models: an overview. In: Kendall, G., Burke, E.K., Petrovic, S., Gendreau, M. (eds.) Multidisciplinary Scheduling: Theory and Applications, pp. 37–53. Springer, Boston (2005). Scholar
  17. 17.
    Mäcker, A., Meyer auf der Heide, F., Pukrop, S.: Approximating weighted completion time for order scheduling with setup times. arXiv e-prints arXiv:1910.08360, October 2019
  18. 18.
    Monma, C.L., Potts, C.N.: On the complexity of scheduling with batch setup times. Oper. Res. 37(5), 798–804 (1989)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ng, C.T., Cheng, T.C.E., Yuan, J.J.: Strong NP-hardness of the single machine multi-operation jobs total completion time scheduling problem. Inf. Process. Lett. 82(4), 187–191 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Smith, W.E.: Various optimizers for single-stage production. Naval Res. Logistics Q. 3(1–2), 59–66 (1956)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Woeginger, G.J.: On the approximability of average completion time scheduling under precedence constraints. Discrete Appl. Math. 131(1), 237–252 (2003)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Alexander Mäcker
    • 1
  • Friedhelm Meyer auf der Heide
    • 1
  • Simon Pukrop
    • 1
    Email author
  1. 1.Heinz Nixdorf Institute and Computer Science DepartmentPaderborn UniversityPaderbornGermany

Personalised recommendations