A garden $G$ is populated by $n\ge 1$ bamboos $b_1, b_2, ..., b_n$ with the respective daily growth rates $h_1 \ge h_2 \ge \dots \ge h_n$. It is assumed that the initial heights of bamboos are zero. The robotic gardener maintaining the garden regularly attends bamboos and trims them to height zero according to some schedule. The Bamboo Garden Trimming Problem (BGT) is to design a perpetual schedule of cuts to maintain the elevation of the bamboo garden as low as possible. The bamboo garden is a metaphor for a collection of machines which have to be serviced, with different frequencies, by a robot which can service only one machine at a time. The objective is to design a perpetual schedule of servicing which minimizes the maximum (weighted) waiting time for servicing. We consider two variants of BGT. In discrete BGT the robot trims only one bamboo at the end of each day. In continuous BGT the bamboos can be cut at any time, however, the robot needs time to move from one bamboo to the next. For discrete BGT, we show tighter approximation algorithms for the case when the growth rates are balanced and for the general case. The former algorithm settles one of the conjectures about the Pinwheel problem. The general approximation algorithm improves on the previous best approximation ratio. For continuous BGT, we propose approximation algorithms which achieve approximation ratios $O(\log \lceil h_1/h_n\rceil)$ and $O(\log n)$.

翻译：一个花园 $G$ 中生长着 $n\ge 1$ 株竹子 $b_1, b_2, ..., b_n$，其每日生长速率分别为 $h_1 \ge h_2 \ge \dots \ge h_n$。假设竹子的初始高度为零。负责维护花园的机器人定期照料竹子，并按照某种调度将其修剪至零高度。竹子花园修剪问题（BGT）旨在设计一种永久的修剪调度，以尽可能保持竹子花园的高度最低。竹子花园是机器集合的隐喻，这些机器需要由一次只能服务一台的机器人以不同频率进行维护。其目标是设计一种永久性的服务调度，以最小化最大（加权）等待服务时间。我们考虑BGT的两种变体。在离散BGT中，机器人每天结束时仅修剪一株竹子。在连续BGT中，竹子可以随时修剪，但机器人需要时间从一株竹子移动到下一株。对于离散BGT，我们针对生长速率平衡的情况和一般情况展示了更紧的近似算法。前者算法解决了关于风车问题的一个猜想。一般近似算法改进了先前的最佳近似比。对于连续BGT，我们提出了近似比分别为 $O(\log \lceil h_1/h_n\rceil)$ 和 $O(\log n)$ 的近似算法。