具有不同紧急程度需求的机器的永久维护
Perpetual maintenance of machines with different urgency requirements
AI总结：
本文研究竹子花园修剪问题（BGT），即为具有不同服务频率需求的机器设计永久维护时间表以最小化最大等待时间，针对离散和连续两种变体分别提出改进的近似算法，并解决了 Pinwheel 问题的一个猜想。
摘要：
一个花园 $G$ 中种植着 $n\ge 1$ 棵竹子 $b_1, b_2, ..., b_n$，它们各自的日生长速率为 $h_1 \ge h_2 \ge \dots \ge h_n$。假设竹子的初始高度为零。维护花园的机器人园丁按照某种时间表定期照料竹子并将其修剪至高度零。竹子花园修剪问题（Bamboo Garden Trimming Problem, BGT）旨在设计一个永久的修剪时间表，以尽可能降低竹子花园的高度。竹子花园是对一组机器的隐喻，这些机器需要由一次只能服务一台机器的机器人以不同频率进行服务。目标是设计一个永久的服务时间表，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，机器人在每天结束时只修剪一棵竹子。在连续 BGT 中，竹子可以在任何时间被切割，但机器人需要时间从一棵竹子移动到下一棵。对于离散 BGT，我们针对生长速率均衡的情形和一般情形给出了更紧的近似算法。前一种算法解决了关于 Pinwheel 问题的一个猜想。一般情形的近似算法改进了此前的最佳近似比。对于连续 BGT，我们提出了近似算法，其近似比分别达到 $O(\log \lceil h_1/h_n\rceil)$ 和 $O(\log n)$。
英文摘要：
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 $O(\log n)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
