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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.16

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

Authors: Felix Höhne and Rob van Stee

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract

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.

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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)

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@InProceedings{hohne_et_al:LIPIcs.APPROX/RANDOM.2023.16,
  author =	{H\"{o}hne, Felix and van Stee, Rob},
  title =	{{A 10/7-Approximation for Discrete Bamboo Garden Trimming and Continuous Trimming on Star Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{16:1--16:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.16},
  URN =		{urn:nbn:de:0030-drops-188417},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.16},
  annote =	{Keywords: bamboo garden trimming, approximation algorithms, scheduling}
}

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

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