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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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Subject Classification

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

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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) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.16

Author Details

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

Supplementary Materials

References

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