Document Open Access Logo

Hardness and Fixed Parameter Tractability for Pinwheel Scheduling Problems

Authors Yusuke Kobayashi , Bingkai Lin

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2025.47.pdf
  • Filesize: 0.75 MB
  • 15 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Pinwheel Scheduling
  • Polynomial-time Solvability
  • Packing and Covering
  • Fixed Parameter Algorithms

Metrics

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

Abstract

In the Pinwheel Packing problem, we are given a set of recurring tasks, each associated with a positive integer a_i for task i. The objective is to select one task to perform each day such that every task i is performed at least once within every a_i consecutive days. The exact computational complexity of this problem, where ∑ 1/a_i = 1, has remained an open question for more than 30 years; in particular, it is still unknown whether the problem is NP-hard. The first contribution of this paper is to show that Pinwheel Packing cannot be solved in polynomial time under a standard complexity assumption, improving upon the hardness result shown by Jacobs and Longo. Additionally, we present fixed-parameter algorithms for variants of Pinwheel Packing, parameterized by the number of tasks.

Cite As Get BibTex

Yusuke Kobayashi and Bingkai Lin. Hardness and Fixed Parameter Tractability for Pinwheel Scheduling Problems. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 47:1-47:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.47

Author Details

Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Bingkai Lin
  • School of Computer Science, Nanjing University, China

Funding

  • Kobayashi, Yusuke: JSPS KAKENHI Grant Numbers JP22H05001 and JP24K02901.

References

  1. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844-856, July 1995. URL: https://doi.org/10.1145/210332.210337.
  2. Paul Balister. Erdős Covering Systems, pages 31-54. London Mathematical Society Lecture Note Series. Cambridge University Press, 2024. Google Scholar
  3. Amotz Bar-Noy, Randeep Bhatia, Joseph Naor, and Baruch Schieber. Minimizing service and operation costs of periodic scheduling. Mathematics of Operations Research, 27(3):518-544, 2002. URL: https://doi.org/10.1287/moor.27.3.518.314.
  4. Amotz Bar-Noy and Richard E. Ladner. Windows scheduling problems for broadcast systems. SIAM Journal on Computing, 32(4):1091-1113, 2003. URL: https://doi.org/10.1137/S009753970240447X.
  5. Amotz Bar-Noy, Richard E. Ladner, and Tami Tamir. Windows scheduling as a restricted version of bin packing. ACM Trans. Algorithms, 3(3):Article 28, 2007. URL: https://doi.org/10.1145/1273340.1273344.
  6. Mee Yee Chan and Francis Chin. Schedulers for larger classes of pinwheel instances. Algorithmica, 9:425-462, 1993. URL: https://doi.org/10.1007/BF01187034.
  7. Peter C. Fishburn and J. C. Lagarias. Pinwheel scheduling: Achievable densities. Algorithmica, 34(1):14-38, 2002. URL: https://doi.org/10.1007/S00453-002-0938-9.
  8. M. R. Garey and D. S. Johnson. Computers and intractability: A guide to the theory of NP-completeness. W.H. Freeman and Co, 2003. Google Scholar
  9. Leszek Gąsieniec, Ralf Klasing, Christos Levcopoulos, Andrzej Lingas, Jie Min, and Tomasz Radzik. Bamboo garden trimming problem (perpetual maintenance of machines with different attendance urgency factors). In Bernhard Steffen, Christel Baier, Mark van den Brand, Johann Eder, Mike Hinchey, and Tiziana Margaria, editors, SOFSEM 2017: Theory and Practice of Computer Science, pages 229-240, Cham, 2017. Springer International Publishing. URL: https://doi.org/10.1007/978-3-319-51963-0_18.
  10. Leszek Gąsieniec, Benjamin Smith, and Sebastian Wild. Towards the 5/6-density conjecture of pinwheel scheduling. In Cynthia A. Phillips and Bettina Speckmann, editors, Proceedings of the Symposium on Algorithm Engineering and Experiments, ALENEX 2022, Alexandria, VA, USA, January 9-10, 2022, pages 91-103. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977042.8.
  11. Leszek Gąsieniec, Tomasz Jurdziński, Ralf Klasing, Christos Levcopoulos, Andrzej Lingas, Jie Min, and Tomasz Radzik. Perpetual maintenance of machines with different urgency requirements. Journal of Computer and System Sciences, 139:103476, 2024. URL: https://doi.org/10.1016/j.jcss.2023.103476.
  12. Robert Holte, Aloysius Mok, Louis Rosier, Igor Tulchinsky, and Donald Varvel. The pinwheel: a real-time scheduling problem. In Proceedings of the Twenty-Second Annual Hawaii International Conference on System Sciences. Volume II: Software Track, volume 2, pages 693-702 vol.2, 1989. URL: https://doi.org/10.1109/HICSS.1989.48075.
  13. Robert Holte, Louis Rosier, Igor Tulchinsky, and Donald Varvel. Pinwheel scheduling with two distinct numbers. Theoretical Computer Science, 100(1):105-135, 1992. URL: https://doi.org/10.1016/0304-3975(92)90365-M.
  14. Bob Hough. Solution of the minimum modulus problem for covering systems. Annals of Mathematics, 181(1):361-382, 2015. URL: https://doi.org/10.4007/annals.2015.181.1.6.
  15. A.P. Huhn and L. Megyesi. On disjoint residue classes. Discrete Mathematics, 41(3):327-330, 1982. URL: https://doi.org/10.1016/0012-365X(82)90030-9.
  16. Tobias Jacobs and Salvatore Longo. A new perspective on the windows scheduling problem, 2014. URL: https://arxiv.org/abs/1410.7237.
  17. Ravi Kannan. Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research, 12(3):415-440, 1987. URL: https://doi.org/10.1287/moor.12.3.415.
  18. Akitoshi Kawamura. Proof of the density threshold conjecture for pinwheel scheduling. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, pages 1816-1819, New York, NY, USA, 2024. Association for Computing Machinery. URL: https://doi.org/10.1145/3618260.3649757.
  19. Akitoshi Kawamura. Perpetual scheduling under frequency constraints. In Shin-ichi Minato, Takeaki Uno, Norihito Yasuda, Takashi Horiyama, Ken-ichi Kawarabayashi, Shigeru Yamashita, and Hirotaka Ono, editors, Algorithmic Foundations for Social Advancement: Recent Progress on Theory and Practice. Springer, 2025. URL: https://doi.org/10.1007/978-981-96-0668-9.
  20. Akitoshi Kawamura, Yosuke Kusano, and Yusuke Kobayashi. Pinwheel covering. In The 14th International Conference on Algorithms and Complexity (CIAC), pages 185-199, 2025. URL: https://doi.org/10.1007/978-3-031-92935-9_12.
  21. Akitoshi Kawamura and Makoto Soejima. Simple strategies versus optimal schedules in multi-agent patrolling. Theoretical Computer Science, 839:195-206, 2020. URL: https://doi.org/10.1016/j.tcs.2020.07.037.
  22. Jan Korst, Emile Aarts, and Jan Karel Lenstra. Scheduling periodic tasks. INFORMS Journal on Computing, 8(4):428-435, 1996. URL: https://doi.org/10.1287/ijoc.8.4.428.
  23. Hendrik W Lenstra Jr. Integer programming with a fixed number of variables. Mathematics of operations research, 8(4):538-548, 1983. URL: https://doi.org/10.1287/moor.8.4.538.
  24. Shun-Shii Lin and Kwei-Jay Lin. A pinwheel scheduler for three distinct numbers with a tight schedulability bound. Algorithmica, 19:411-426, 1997. URL: https://doi.org/10.1007/PL00009181.
  25. Al Mok, Louis Rosier, Igor Tulchinsky, and Donald Varvel. Algorithms and complexity of the periodic maintenance problem. Microprocessing and Microprogramming, 27(1):657-664, 1989. Fifteenth EUROMICRO Symposium on Microprocessing and Microprogramming. URL: https://doi.org/10.1016/0165-6074(89)90128-2.
  26. Moni Naor, Leonard J Schulman, and Aravind Srinivasan. Splitters and near-optimal derandomization. In Proceedings of IEEE 36th Annual Foundations of Computer Science, pages 182-191. IEEE, 1995. URL: https://doi.org/10.1109/SFCS.1995.492475.
  27. Victor Reis and Thomas Rothvoss. The subspace flatness conjecture and faster integer programming. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 974-988, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00060.
  28. Zhi-Wei Sun. On disjoint residue classes. Discrete Mathematics, 104(3):321-326, 1992. URL: https://doi.org/10.1016/0012-365X(92)90454-N.
  29. Martijn van Ee. A 12/7-approximation algorithm for the discrete bamboo garden trimming problem. Operations Research Letters, 49(5):645-649, 2021. URL: https://doi.org/10.1016/j.orl.2021.07.001.
  30. W.D Wei and C.L Liu. On a periodic maintenance problem. Operations Research Letters, 2(2):90-93, 1983. URL: https://doi.org/10.1016/0167-6377(83)90044-5.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact