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

Authors Matthias Kaul , Matthias Mnich , Hendrik Molter

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

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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.

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

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.

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