Document Open Access Logo

A Computer-Assisted Proof of the Optimal Density Bound for Pinwheel Covering

Authors Akitoshi Kawamura , Yusuke Kobayashi

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2026.83.pdf
  • Filesize: 0.64 MB
  • 7 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • pinwheel scheduling
  • pinwheel covering
  • density threshold

Metrics

Abstract

In the covering version of the pinwheel scheduling problem, a daily task must be assigned to agents under the constraint that agent i can perform the task at most once in any a_i-day interval. In this paper, we determine the optimal constant α^* = 1.264… such that every instance with ∑_i 1/a_i ≥ α^* is schedulable. This resolves an open problem posed by Kawamura and Soejima (2020). Our proof combines Kawamura’s (2026) techniques for the packing version with new mathematical insights to reduce the analysis to a finite set of instances, which are then verified through an exhaustive computer-aided search that draws on ideas from Gąsieniec, Smith, and Wild (2022). The same result was obtained independently by Mishra (2026).

Cite As Get BibTex

Akitoshi Kawamura and Yusuke Kobayashi. A Computer-Assisted Proof of the Optimal Density Bound for Pinwheel Covering. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 83:1-83:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.83

Author Details

Akitoshi Kawamura
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan

Funding

  • Kawamura, Akitoshi: Support Center for Advanced Telecommunications Technology Research (SCAT); Kyoto University Foundation.
  • Kobayashi, Yusuke: JSPS KAKENHI Grant Numbers JP22H05001, 24K02901, 26K21777; JST ERATO Grant Number JPMJER2301.

Supplementary Materials

References

  1. M. Y. Chan and Francis Chin. Schedulers for larger classes of pinwheel instances. Algorithmica, 9:425-462, 1993. URL: https://doi.org/10.1007/BF01187034.
  2. 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.
  3. Leszek Gąsieniec, Benjamin Smith, and Sebastian Wild. Towards the 5/6-density conjecture of pinwheel scheduling. In 2022 Proceedings of the Symposium on Algorithm Engineering and Experiments (ALENEX), pages 91-103, 2022. URL: https://doi.org/10.1137/1.9781611977042.8.
  4. R. Holte, A. Mok, L. Rosier, I. Tulchinsky, and D. 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, 1989. URL: https://doi.org/10.1109/HICSS.1989.48075.
  5. Akitoshi Kawamura. Pinwheel Schedulers. Software, version v1.0.0., (visited on 2026-08-12). URL: https://github.com/a7kawamura/pinwheel_solver
    Software Heritage Logo archived version
    full metadata available at: https://doi.org/10.4230/artifacts.27639
  6. 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. Google Scholar
  7. Akitoshi Kawamura. Proof of the density threshold conjecture for pinwheel scheduling. Proceedings of the National Academy of Sciences, 123(32):e2530214123, 2026. URL: https://doi.org/10.1073/pnas.2530214123.
  8. Akitoshi Kawamura and Yusuke Kobayashi. A computer-assisted proof of the optimal density bound for pinwheel covering, 2025. URL: https://doi.org/10.48550/arXiv.2510.06533.
  9. Akitoshi Kawamura, Yusuke Kobayashi, and Yosuke Kusano. Pinwheel covering. In Irene Finocchi and Loukas Georgiadis, editors, Algorithms and Complexity - 14th International Conference, CIAC 2025, Rome, Italy, June 10-12, 2025, Proceedings, Part II, volume 15680 of Lecture Notes in Computer Science, pages 185-199. Springer, 2025. URL: https://doi.org/10.1007/978-3-031-92935-9_12.
  10. 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.
  11. Yusuke Kobayashi, Bingkai Lin, and Joseph Swernofsky. Hardness and fixed parameter tractability for pinwheel scheduling problems. Theoretical Computer Science, 1077:115998, 2026. URL: https://doi.org/10.1016/j.tcs.2026.115998.
  12. 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.
  13. Ahan Mishra. An optimal density bound for discretized point patrolling. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2846-2875, 2026. URL: https://doi.org/10.1137/1.9781611978971.105.
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
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact