Document Open Access Logo

Single-Machine Scheduling to Minimize the Number of Tardy Jobs with Release Dates

Authors Matthias Kaul , Matthias Mnich , Hendrik Molter

PDF
Thumbnail PDF

File

LIPIcs.IPEC.2024.19.pdf
  • Filesize: 0.78 MB
  • 15 pages

Document Identifiers

Author Details

Matthias Kaul
  • Universität Bonn, Bonn, Germany
Matthias Mnich
  • Hamburg University of Technology, Institute for Algorithms and Complexity, Hamburg, Germany
Hendrik Molter
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

  • Mnich, Matthias: Partially supported by DFG project MN 59/4-1.
  • Molter, Hendrik: Supported by the European Union’s Horizon Europe research and innovation programme under grant agreement 949707.

Acknowledgements

We wish to thank Danny Hermelin and Dvir Shabtay for fruitful discussions that led to some of the results of this work.

Cite As Get BibTex

Matthias Kaul, Matthias Mnich, and Hendrik Molter. Single-Machine Scheduling to Minimize the Number of Tardy Jobs with Release Dates. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.19

Abstract

We study the fundamental scheduling problem 1|r_j|∑ w_j U_j: schedule a set of n jobs with weights, processing times, release dates, and due dates on a single machine, such that each job starts after its release date and we maximize the weighted number of jobs that complete execution before their due date. Problem 1|r_j|∑ w_j U_j generalizes both Knapsack and Partition, and the simplified setting without release dates was studied by Hermelin et al. [Annals of Operations Research, 2021] from a parameterized complexity viewpoint.
Our main contribution is a thorough complexity analysis of 1|r_j|∑ w_j U_j in terms of four key problem parameters: the number p_# of processing times, the number w_# of weights, the number d_# of due dates, and the number r_# of release dates of the jobs. 1|r_j|∑ w_j U_j is known to be weakly para-NP-hard even if w_#+d_#+r_# is constant, and Heeger and Hermelin [ESA, 2024] recently showed (weak) 𝖶[1]-hardness parameterized by p_# or w_# even if r_# is constant.
Algorithmically, we show that 1|r_j|∑ w_j U_j is fixed-parameter tractable parameterized by p_# combined with any two of the remaining three parameters w_#, d_#, and r_#. We further provide pseudo-polynomial XP-time algorithms for parameter r_# and d_#. To complement these algorithms, we show that 1|r_j|∑ w_j U_j is (strongly) 𝖶[1]-hard when parameterized by d_#+r_# even if w_# is constant. Our results provide a nearly complete picture of the complexity of 1|r_j|∑ w_j U_j for p_#, w_#, d_#, and r_# as parameters, and extend those of Hermelin et al. [Annals of Operations Research, 2021] for the problem 1||∑ w_j U_j without release dates.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Scheduling algorithms
Keywords
  • Scheduling
  • Release Dates
  • Fixed-Parameter Tractability

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Esther M. Arkin and Ellen B. Silverberg. Scheduling jobs with fixed start and end times. Discrete Appl. Math., 18(1):1-8, 1987. URL: https://doi.org/10.1016/0166-218X(87)90037-0.
  2. Philippe Baptiste, Peter Brucker, Sigrid Knust, and Vadim G Timkovsky. Ten notes on equal-processing-time scheduling: at the frontiers of solvability in polynomial time. Quarterly J. Belgian, French and Ital. Oper. Res. Soc., 2(2):111-127, 2004. URL: https://doi.org/10.1007/s10288-003-0024-4.
  3. Karl Bringmann, Nick Fischer, Danny Hermelin, Dvir Shabtay, and Philip Wellnitz. Faster minimization of tardy processing time on a single machine. Algorithmica, 84(5):1341-1356, 2022. URL: https://doi.org/10.1007/S00453-022-00928-W.
  4. Jan Elffers and Mathijs de Weerdt. Scheduling with two non-unit task lengths is NP-complete. Technical report, 2014. URL: https://doi.org/10.48550/arXiv.1412.3095.
  5. Michael R. Fellows, Serge Gaspers, and Frances A. Rosamond. Parameterizing by the number of numbers. Theory Comput. Syst., 50(4):675-693, 2012. URL: https://doi.org/10.1007/S00224-011-9367-Y.
  6. Nick Fischer and Leo Wennmann. Minimizing tardy processing time on a single machine in near-linear time. Technical report, 2024. URL: https://doi.org/10.48550/arXiv.2402.13357.
  7. M. R. Garey, D. S. Johnson, B. B. Simons, and R. E. Tarjan. Scheduling unit–time tasks with arbitrary release times and deadlines. SIAM J. Comput., 10(2):256-269, 1981. URL: https://doi.org/10.1137/0210018.
  8. Ronald L. Graham. Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math., 17(2):416-429, 1969. URL: https://doi.org/10.1137/0117039.
  9. Klaus Heeger and Danny Hermelin. Minimizing the weighted number of tardy jobs is 𝖶[1]-hard. In Proc. ESA 2024, volume 308 of Leibniz Int. Proc. Informatics, pages 68:1-68:14, 2024. URL: https://doi.org/10.4230/LIPICS.ESA.2024.68.
  10. Klaus Heeger, Danny Hermelin, George B. Mertzios, Hendrik Molter, Rolf Niedermeier, and Dvir Shabtay. Equitable scheduling on a single machine. J. Sched., 26(2):209-225, 2023. URL: https://doi.org/10.1007/S10951-022-00754-6.
  11. Klaus Heeger and Hendrik Molter. Minimizing the number of tardy jobs with uniform processing times on parallel machines. Technical report, 2024. URL: https://doi.org/10.48550/arXiv.2404.14208.
  12. Danny Hermelin, Yuval Itzhaki, Hendrik Molter, and Dvir Shabtay. On the parameterized complexity of interval scheduling with eligible machine sets. J. Comput. Syst. Sci., page 103533, 2024. URL: https://doi.org/10.1016/J.JCSS.2024.103533.
  13. Danny Hermelin, Shlomo Karhi, Michael L. Pinedo, and Dvir Shabtay. New algorithms for minimizing the weighted number of tardy jobs on a single machine. Ann. Oper. Res., 298(1):271-287, 2021. URL: https://doi.org/10.1007/S10479-018-2852-9.
  14. Danny Hermelin, Hendrik Molter, and Dvir Shabtay. Minimizing the weighted number of tardy jobs via (max,+)-convolutions. INFORMS J. Comput., 36:836-848, 2024. URL: https://doi.org/10.1287/IJOC.2022.0307.
  15. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85-103. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  16. Matthias Kaul, Matthias Mnich, and Hendrik Molter. Single-machine scheduling to minimize the number of tardy jobs with release dates. Technical report, 2024. URL: https://doi.org/10.48550/arXiv.2408.12967.
  17. Kim-Manuel Klein, Adam Polak, and Lars Rohwedder. On minimizing tardy processing time, max-min skewed convolution, and triangular structured ILPs. In Proc. SODA 2023, pages 2947-2960, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH112.
  18. Sven O. Krumke, Clemens Thielen, and Stephan Westphal. Interval scheduling on related machines. Comput. Oper. Res., 38(12):1836-1844, 2011. URL: https://doi.org/10.1016/J.COR.2011.03.001.
  19. Eugene L. Lawler and James M. Moore. A functional equation and its application to resource allocation and sequencing problems. Mgmt. Sci., 16(1):77-84, 1969. URL: https://doi.org/10.1287/mnsc.16.1.77.
  20. Jan K. Lenstra, A.H.G. Rinnooy Kan, and Peter Brucker. Complexity of machine scheduling problems. Ann. Discrete Math., 1:343-362, 1977. URL: https://doi.org/10.1016/S0167-5060(08)70743-X.
  21. Hendrik W Lenstra Jr. Integer programming with a fixed number of variables. Math. Oper. Res., 8(4):538-548, 1983. URL: https://doi.org/10.1287/MOOR.8.4.538.
  22. Matthias Mnich and René van Bevern. Parameterized complexity of machine scheduling: 15 open problems. Comput. Oper. Res., 100:254-261, 2018. URL: https://doi.org/10.1016/J.COR.2018.07.020.
  23. Matthias Mnich and Andreas Wiese. Scheduling and fixed-parameter tractability. Math. Prog., 154(1):533-562, 2015. URL: https://doi.org/10.1007/S10107-014-0830-9.
  24. James M. Moore. An n job, one machine sequencing algorithm for minimizing the number of late jobs. Mgmt. Sci., 15:102-109, 1968. URL: https://doi.org/10.1287/mnsc.15.1.102.
  25. Michael Pinedo. Scheduling: Theory, Algorithms, and Systems, 5th Edition. Springer, 2016. Google Scholar
  26. Baruch Schieber and Pranav Sitaraman. Quick minimization of tardy processing time on a single machine. In Proc. WADS 2023, volume 14079 of Lecture Notes Comput. Sci., pages 637-643, 2023. URL: https://doi.org/10.1007/978-3-031-38906-1_42.
  27. Shao Chin Sung and Milan Vlach. Maximizing weighted number of just-in-time jobs on unrelated parallel machines. J. Sched., 8(5):453-460, 2005. URL: https://doi.org/10.1007/S10951-005-2863-7.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact