A 10/7-Approximation for Discrete Bamboo Garden Trimming and Continuous Trimming on Star Graphs

Authors Felix Höhne, Rob van Stee

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Felix Höhne
  • Fraunhofer Institute for Industrial Mathematics ITWM, Kaiserslautern, Germany
Rob van Stee
  • University of Siegen, Germany

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Felix Höhne and Rob van Stee. A 10/7-Approximation for Discrete Bamboo Garden Trimming and Continuous Trimming on Star Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


In the discrete bamboo garden trimming problem we are given n bamboo that grow at rates v_1,… ,v_n per day. Each day a robotic gardener cuts down one bamboo to height 0. The goal is to find a schedule that minimizes the height of the tallest bamboo that ever exists. We present a 10/7-approximation algorithm that is based on a reduction to the pinwheel problem. This is consistent with the approach of earlier algorithms, but some new techniques are used that lead to a better approximation ratio. We also consider the continuous version of the problem where the gardener travels in a metric space between plants and cuts down a plant each time he reaches one. We show that on the star graph the previously proposed algorithm Reduce-Fastest is a 6-approximation and the known Deadline-Driven Strategy is a (3+2√2)-approximation. The Deadline-Driven Strategy is also a (9+2√5)-approximation on star graphs with multiple plants on each branch.

Subject Classification

ACM Subject Classification
  • Theory of computation → Scheduling algorithms
  • bamboo garden trimming
  • approximation algorithms
  • scheduling


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  1. Michael A. Bender, Sándor P. Fekete, Alexander Kröller, Vincenzo Liberatore, Joseph S. B. Mitchell, Valentin Polishchuk, and Jukka Suomela. The minimum backlog problem. Theor. Comput. Sci., 605:51-61, 2015. URL: https://doi.org/10.1016/j.tcs.2015.08.027.
  2. Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Giacomo Scornavacca. Cutting bamboo down to size. In Martin Farach-Colton, Giuseppe Prencipe, and Ryuhei Uehara, editors, 10th International Conference on Fun with Algorithms, FUN 2021, May 30 to June 1, 2021, Favignana Island, Sicily, Italy, volume 157 of LIPIcs, pages 5:1-5:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.FUN.2021.5.
  3. Marijke H. L. Bodlaender, Cor A. J. Hurkens, Vincent J. J. Kusters, Frank Staals, Gerhard J. Woeginger, and Hans Zantema. Cinderella versus the wicked stepmother. In Jos C. M. Baeten, Thomas Ball, and Frank S. de Boer, editors, Theoretical Computer Science - 7th IFIP TC 1/WG 2.2 International Conference, TCS 2012, Amsterdam, The Netherlands, September 26-28, 2012. Proceedings, volume 7604 of Lecture Notes in Computer Science, pages 57-71. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-33475-7_5.
  4. Mee Yee Chan and Francis Y. L. Chin. General schedulers for the pinwheel problem based on double-integer reduction. IEEE Trans. Computers, 41(6):755-768, 1992. URL: https://doi.org/10.1109/12.144627.
  5. Marek Chrobak, János Csirik, Csanád Imreh, John Noga, Jirí Sgall, and Gerhard J. Woeginger. The buffer minimization problem for multiprocessor scheduling with conflicts. In Fernando Orejas, Paul G. Spirakis, and Jan van Leeuwen, editors, Automata, Languages and Programming, 28th International Colloquium, ICALP 2001, Crete, Greece, July 8-12, 2001, Proceedings, volume 2076 of Lecture Notes in Computer Science, pages 862-874. Springer, 2001. URL: https://doi.org/10.1007/3-540-48224-5_70.
  6. Wei Ding. A branch-and-cut approach to examining the maximum density guarantee for pinwheel schedulability of low-dimensional vectors. Real Time Syst., 56(3):293-314, 2020. URL: https://doi.org/10.1007/s11241-020-09349-w.
  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. Leszek Gasieniec, 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 - 43rd International Conference on Current Trends in Theory and Practice of Computer Science, Limerick, Ireland, January 16-20, 2017, Proceedings, volume 10139 of Lecture Notes in Computer Science, pages 229-240. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-51963-0_18.
  9. Leszek Gasieniec, 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.
  10. Robert Holte, Aloysius Mok, Louis Rosier, Igor Tulchinsky, and Donald Varvel. Pinwheel: a real-time scheduling problem. In Proceedings of the Hawaii International Conference on System Science, pages 693-702 vol.2, 1989. URL: https://doi.org/10.1109/HICSS.1989.48075.
  11. Robert Holte, Louis E. Rosier, Igor Tulchinsky, and Donald A. Varvel. Pinwheel scheduling with two distinct numbers. Theor. Comput. Sci., 100(1):105-135, 1992. URL: https://doi.org/10.1016/0304-3975(92)90365-M.
  12. John Kuszmaul. Bamboo trimming revisited: Simple algorithms can do well too. CoRR, abs/2201.07350, 2022. URL: https://arxiv.org/abs/2201.07350.
  13. William Kuszmaul. Achieving optimal backlog in the vanilla multi-processor cup game. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1558-1577. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.96.
  14. Martijn van Ee. A 12/7-approximation algorithm for the discrete bamboo garden trimming problem. CoRR, abs/2004.11731, 2020. URL: https://arxiv.org/abs/2004.11731.