竹园修剪问题的改进风车算法
An enhanced pinwheel algorithm for the bamboo garden trimming problem
摘要：
在竹园修剪问题（BGT）中，有一个由n棵竹子b（1），b（2），.，b（n）$，其日增长率h（1）>= h（2）>=. n = h（n）.我们假设竹子的初始高度为零。一个园丁负责竹子，并根据一些时间表将它们修剪到零高度。目的是设计一个永久的修剪时间表，以保持竹园的高度尽可能低。我们考虑所谓的离散BGT变体，其中园丁被允许在每天结束时只修剪一根竹子。对于离散BGT，目前最先进的近似算法利用BGT和经典的风车调度问题之间的关系，并提供了一个解决方案，保证2-近似比。我们提出了一种替代的风车调度算法，当sum h（j）> > h（1）时，近似比收敛到12/7。此外，我们表明，所提出的算法的近似比从来没有超过32000/16947约1.888。这是第一个达到严格低于19/10的比率的算法。
英文摘要：
In the Bamboo Garden Trimming Problem (BGT), there is a garden populated by n bamboos b(1), b(2), ... , b(n)$ with daily growth rates h(1) >= h(2) >= ... >= h(n). We assume that the initial heights of bamboos are zero. A gardener is in charge of the bamboos and trims them to height zero according to some schedule. The objective is to design a perpetual schedule of trimming so as to maintain the height of the bamboo garden as low as possible. We consider the so-called discrete BGT variant, where the gardener is allowed to trim only one bamboo at the end of each day. For discrete BGT, the current state-of-the-art approximation algorithm exploits the relationship between BGT and the classical Pinwheel scheduling problem and provides a solution that guarantees a 2-approximation ratio. We propose an alternative Pinwheel scheduling algorithm with approximation ratio converging to 12/7 when sum h(j) > > h(1). Also, we show that the approximation ratio of the proposed algorithm never exceeds 32000/16947 approximately 1.888. This is the first algorithm reaching a ratio strictly inferior to 19/10.
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