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Minimizing the Weighted Number of Tardy Jobs Is W[1]-Hard

Authors Klaus Heeger , Danny Hermelin

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Combinatorial optimization
Keywords
  • single-machine scheduling
  • number of different weights
  • number of different processing times

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Abstract

We consider the 1∣∣∑ w_jU_j problem, the problem of minimizing the weighted number of tardy jobs on a single machine. This problem is one of the most basic and fundamental problems in scheduling theory, with several different applications both in theory and practice. Using a reduction from the Multicolored Clique problem, we prove that 1∣∣∑ w_jU_j is W[1]-hard with respect to the number p_# of different processing times in the input, as well as with respect to the number w_# of different weights in the input. This, along with previous work, provides a complete picture for 1∣∣∑ w_jU_j from the perspective of parameterized complexity, as well as almost tight complexity bounds for the problem under the Exponential Time Hypothesis (ETH).

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Klaus Heeger and Danny Hermelin. Minimizing the Weighted Number of Tardy Jobs Is W[1]-Hard. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 68:1-68:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.68

Author Details

Klaus Heeger
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Danny Hermelin
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

Supported by the ISF, grant No. 1070/20.

References

  1. Muminu O. Adamu and Aderemi O. Adewumi. A survey of single machine scheduling to minimize weighted number of tardy jobs. Journal of Industrial and Management Optimization, 10(1):219-241, 2014. Google Scholar
  2. 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. Google Scholar
  3. Marek Cygan, Marcin Mucha, Karol Wegrzycki, and Michal Wlodarczyk. On problems equivalent to (min, +)-convolution. ACM Transactions on Algorithms, 15(1):14:1-14:25, 2019. Google Scholar
  4. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Monographs in Computer Science. Springer, 1999. Google Scholar
  5. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  6. Michael R. Fellows, Serge Gaspers, and Frances A. Rosamond. Parameterizing by the number of numbers. Theory of Computing Systems, 50(4):675-693, 2012. Google Scholar
  7. Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theoretical Computer Science, 410(1):53-61, 2009. Google Scholar
  8. Michel X. Goemans and Thomas Rothvoss. Polynomiality for bin packing with a constant number of item types. Journal of the ACM, 67(6):38:1-38:21, 2020. URL: https://doi.org/10.1145/3421750.
  9. Ronald L. Graham. Bounds on multiprocessing timing anomalies. SIAM Journal on Applied Mathematics, 17(2):416-429, 1969. Google Scholar
  10. Klaus Heeger and Danny Hermelin. Minimizing the weighted number of tardy jobs is W[1]-hard. CoRR, abs/2401.01740, 2024. URL: https://doi.org/10.48550/arXiv.2401.01740.
  11. Danny Hermelin, Shlomo Karhi, Michael L. Pinedo, and Dvir Shabtay. New algorithms for minimizing the weighted number of tardy jobs on a single machine. Annals of Operations Research, 298(1):271-287, 2021. Google Scholar
  12. Danny Hermelin, Matthias Mnich, and Simon Omlor. Single machine batch scheduling to minimize the weighted number of tardy jobs. CoRR, abs/1911.12350, 2019. Google Scholar
  13. Danny Hermelin, Hendrik Molter, and Dvir Shabtay. Minimizing the weighted number of tardy jobs via (max, +)-convolutions. INFORMS Journal on Computing - to appear, 2024. Google Scholar
  14. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. Google Scholar
  15. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85-103. Springer, 1972. Google Scholar
  16. Kim-Manuel Klein, Adam Polak, and Lars Rohwedder. On minimizing tardy processing time, max-min skewed convolution, and triangular structured ILPs. In Proc. of the 34th ACM-SIAM Symposium On Discrete Algorithms, SODA 2023, pages 2947-2960, 2023. Google Scholar
  17. Eugene L. Lawler and James M. Moore. A functional equation and its application to resource allocation and sequencing problems. Management Science, 16(1):77-84, 1969. Google Scholar
  18. Hendrik W Lenstra Jr. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8(4):538-548, 1983. Google Scholar
  19. Matthias Mnich and René van Bevern. Parameterized complexity of machine scheduling: 15 open problems. Computers & Operations Research, 100:254-261, 2018. Google Scholar
  20. James M. Moore. An n job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15(1):102-109, 1968. Google Scholar
  21. Jon M. Peha. Heterogeneous-criteria scheduling: Minimizing weighted number of tardy jobs and weighted completion time. Computers and Operations Research, 22(10):1089-1100, 1995. Google Scholar
  22. Krzysztof Pietrzak. On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. Journal of Computer and System Sciences, 67(4):757-771, 2003. URL: https://doi.org/10.1016/S0022-0000(03)00078-3.
  23. Victor Reis and Thomas Rothvoss. The subspace flatness conjecture and faster integer programming. In Proc. of the 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, pages 974-988, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00060.
  24. Sartaj K. Sahni. Algorithms for scheduling independent tasks. Journal of the ACM, 23(1):116-127, 1976. Google Scholar
  25. Baruch Schieber and Pranav Sitaraman. Quick minimization of tardy processing time on a single machine. In Proc. of the 18th international Workshop on Algorithms and Data Structures, WADS 2023, pages 637-643, 2023. Google Scholar
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