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Minimizing the Number of Tardy Jobs with Uniform Processing Times on Parallel Machines (2404.14208v1)

Published 22 Apr 2024 in cs.DS

Abstract: In this work, we study the computational (parameterized) complexity of $P \mid r_j, p_j=p \mid \sum_j w_j U_j$. Here, we are given $m$ identical parallel machines and $n$ jobs with equal processing time, each characterized by a release date, a due date, and a weight. The task is to find a feasible schedule, that is, an assignment of the jobs to starting times on machines, such that no job starts before its release date and no machine processes several jobs at the same time, that minimizes the weighted number of tardy jobs. A job is considered tardy if it finishes after its due date. Our main contribution is showing that $P \mid r_j, p_j=p \mid \sum_j U_j$ (the unweighted version of the problem) is NP-hard and W[2]-hard when parameterized by the number of machines. The former resolves an open problem in Note 2.1.19 by Kravchenko and Werner [Journal of Scheduling, 2011] and Open Problem 2 by Sgall [ESA, 2012], and the latter resolves Open Problem 7 by Mnich and van Bevern [Computers & Operations Research, 2018]. Furthermore, our result shows that the known XP-algorithm for $P \mid r_j, p_j=p \mid \sum_j w_j U_j$ parameterized by the number of machines is optimal from a classification standpoint. On the algorithmic side, we provide alternative running time bounds for the above-mentioned known XP-algorithm. Our analysis shows that $P \mid r_j, p_j=p \mid \sum_j w_j U_j$ is contained in XP when parameterized by the processing time, and that it is contained in FPT when parameterized by the combination of the number of machines and the processing time. Finally, we give an FPT-algorithm for $P \mid r_j, p_j=p \mid \sum_j w_j U_j$ parameterized by the number of release dates or the number of due dates. With this work, we lay out the foundation for a systematic study of the parameterized complexity of $P \mid r_j, p_j=p \mid \sum_j w_j U_j$.

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References (34)
  1. Scheduling jobs with fixed start and end times. Discrete Applied Mathematics, 18(1):1–8, 1987.
  2. Philippe Baptiste. Scheduling equal-length jobs on identical parallel machines. Discrete Applied Mathematics, 103(1-3):21–32, 2000.
  3. Ten notes on equal-processing-time scheduling: at the frontiers of solvability in polynomial time. Quarterly Journal of the Belgian, French and Italian Operations Research Societies, 2(2):111–127, 2004.
  4. Faster minimization of tardy processing time on a single machine. Algorithmica, 84(5):1341–1356, 2022.
  5. Scheduling jobs with equal processing times and time windows on identical parallel machines. Journal of Scheduling, 11(4):229–237, 2008.
  6. Parameterized analysis of bribery in challenge the champ tournaments. CoRR, abs/2403.17587, 2024. URL: https://doi.org/10.48550/arXiv.2403.17587.
  7. Parameterized Algorithms. Springer, 2015.
  8. Enumerative lattice algorithms in any norm via M𝑀Mitalic_M-ellipsoid coverings. In Proceedings of the 52nd IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 580–589. IEEE, 2011.
  9. George Bernard Dantzig. Linear inequalities and related systems. Number 38. Princeton University Press, 1956.
  10. Parameterized Complexity. Springer, 1999.
  11. Fundamentals of Parameterized Complexity. Springer, 2013.
  12. Parameterized Complexity Theory, volume XIV of Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006.
  13. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979.
  14. Ronald L. Graham. Bounds on multiprocessing timing anomalies. SIAM Journal on Applied Mathematics, 17(2):416–429, 1969.
  15. 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, arXiv:2401.01740, doi:10.48550/ARXIV.2401.01740.
  16. Equitable scheduling on a single machine. Journal of Scheduling, 26(2):209–225, 2023.
  17. On the parameterized complexity of interval scheduling with eligible machine sets. Journal of Computer and System Sciences, page 103533, 2024.
  18. New algorithms for minimizing the weighted number of tardy jobs on a single machine. Annals of Operations Research, 298(1):271–287, 2021.
  19. Minimizing the weighted number of tardy jobs via (max,+)-convolutions. INFORMS Journal on Computing, 2023.
  20. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, pages 85–103. Springer, 1972.
  21. Parallel machine problems with equal processing times: a survey. Journal of Scheduling, 14:435–444, 2011.
  22. Interval scheduling on related machines. Computers & Operations Research, 38(12):1836–1844, 2011.
  23. Harold W Kuhn. The hungarian method for the assignment problem. Naval research logistics quarterly, 2(1-2):83–97, 1955.
  24. A functional equation and its application to resource allocation and sequencing problems. Management Science, 16(1):77–84, 1969.
  25. Hendrik W. Lenstra. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8:538–548, 1983.
  26. Complexity of machine scheduling problems. 1:343–362, 1977.
  27. Bicriterion scheduling with equal processing times on a batch processing machine. Computers & Operations Research, 36(1):110–118, 2009.
  28. Matthias Mnich and René van Bevern. Parameterized complexity of machine scheduling: 15 open problems. Computers & Operations Research, 100:254–261, 2018.
  29. James M. Moore. An n𝑛nitalic_n job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15:102–109, 1968.
  30. Michael L. Pinedo. Scheduling: Theory, Algorithms, and Systems, 5th Edition. Springer, 2016.
  31. Jirí Sgall. Open problems in throughput scheduling. In Proceedings of the 20th Annual European Symposium on Algorithms (ESA), volume 7501 of Lecture Notes in Computer Science, pages 2–11. Springer, 2012.
  32. Barbara B. Simons. Multiprocessor scheduling of unit-time jobs with arbitrary release times and deadlines. SIAM Journal on Computing, 12(2):294–299, 1983.
  33. A fast algorithm for multiprocessor scheduling of unit-length jobs. SIAM Journal on Computing, 18(4):690–710, 1989.
  34. Maximizing weighted number of just-in-time jobs on unrelated parallel machines. Journal of Scheduling, 8(5):453–460, 2005.
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