A 12/7-approximation algorithm for the discrete Bamboo Garden Trimming problem

Ee, Martijn van

doi:10.1016/j.orl.2021.07.001

Cited by 7 publications

(2 citation statements)

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“…Gasieniec et al provide a 2-approximation. The approximation ratio has subsequently been improved to 1.888 [16] and 12 7 [28]. The problem was also considered by Bilò et al [4], who studied the approximation ratio of simpler algorithms, like cutting down the largest bamboo every day.…”

Section: Related Workmentioning

confidence: 99%

Approximation Algorithms for Replenishment Problems with Fixed Turnover Times

Bosman¹,

Jiao

et al. 2022

Algorithmica

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We introduce and study a class of optimization problems we call replenishment problems with fixed turnover times: a very natural model that has received little attention in the literature. Clients with capacity for storing a certain commodity are located at various places; at each client the commodity depletes within a certain time, the turnover time, which is constant but can vary between locations. Clients should never run empty. The natural feature that makes this problem interesting is that we may schedule a replenishment (well) before a client becomes empty, but then the next replenishment will be due earlier also. This added workload needs to be balanced against the cost of routing vehicles to do the replenishments. In this paper, we focus on the aspect of minimizing routing costs. However, the framework of recurring tasks, in which the next job of a task must be done within a fixed amount of time after the previous one is much more general and gives an adequate model for many practical situations. Note that our problem has an infinite time horizon. However, it can be fully characterized by a compact input, containing only the location of each client and a turnover time. This makes determining its computational complexity highly challenging and indeed it remains essentially unresolved. We study the problem for two objectives: min–avg minimizes the average tour cost and min–max minimizes the maximum tour cost over all days. For min–max we derive a logarithmic factor approximation for the problem on general metrics and a 6-approximation for the problem on trees, for which we have a proof of NP-hardness. For min–avg we present a logarithmic factor approximation on general metrics, a 2-approximation for trees, and a pseudopolynomial time algorithm for the line. Many intriguing problems remain open.

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Section: Related Workmentioning

confidence: 99%

Approximation Algorithms for Replenishment Problems with Fixed Turnover Times

Bosman¹,

Jiao

et al. 2022

Algorithmica

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“…It is known that no bamboo trimming algorithm can guarantee a backlog less than 2, as it is possible to achieve backlog at least 2 − 2ε against any bamboo trimming algorithm with two bamboo that have fill rates 1 − ε and ε [11]. Recent work has yielded complex pinwheel algorithms [18] that achieve backlog 2, and are thus optimal in terms of the worst-case backlog; there has also been effort to extend the guarantees of these algorithms to achieve strong competitive ratios for cases where backlog less than 2 is possible [12,33].…”

Section: Introductionmentioning

confidence: 99%

Bamboo Trimming Revisited: Simple Algorithms Can Do Well Too

Kuszmaul

2022

Preprint

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The bamboo trimming problem considers n bamboo with growth rates h 1 , h 2 , . . . , h n satisfying i h i = 1. During a given unit of time, each bamboo grows by h i , and then the bamboo-trimming algorithm gets to trim one of the bamboo back down to height zero. The goal is to minimize the height of the tallest bamboo, also known as the backlog. The bamboo trimming problem is closely related to many scheduling problems, and can be viewed as a variation of the widely-studied fixed-rate cup game, but with constant-factor resource augmentation.Past work has given sophisticated pinwheel algorithms that achieve the optimal backlog of 2 in the bamboo trimming problem. It has remained an open question, however, whether there exists a simple algorithm with the same guarantee-recent work has devoted considerable theoretical and experimental effort to answering this question. Two algorithms, in particular, have appeared as natural candidates: the Reduce-Max algorithm (which always cuts the tallest bamboo), and the Reduce-Fastest(x) algorithm (which cuts the fastest-growing bamboo out of those that have at least some height x). It is conjectured that Reduce-Max and Reduce-Fastest(1) both achieve backlog 2.This paper improves the bounds for both Reduce-Fastest and Reduce-Max. Among other results, we show that the exact optimal backlog for Reduce-Fastest(x) is x + 1 for all x ≥ 2 (this proves a conjecture of D'Emidio, Di Stefano, and Navarra in the case of x = 2), and we show that Reduce-Fastest(1) does not achieve backlog 2 (this disproves a conjecture of D'Emidio, Di Stefano, and Navarra).Finally, we show that there is a different algorithm, which we call the Deadline-Driven Strategy, that is both very simple and achieves the optimal backlog of 2. This resolves the question as to whether there exists a simple worst-case optimal algorithm for the bamboo-trimming problem.

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