Bamboo Garden Trimming Problem: Priority Schedulings

Abstract

The paper deals with the Bamboo Garden Trimming (BGT) problem introduced in [Gąsieniec et al., SOFSEM’17]. The problem is difficult to solved due to its close relationship to Pinwheel scheduling. The garden with n bamboos is an analogue of a system of n machines that have to be attended (e.g., serviced) with different frequencies. During each day, bamboo b i grows an extra height h i , for i = 1 , ⋯ , n and, on the conclusion of the day, at most one bamboo has its entire height cut.The goal is to design a perpetual schedule of cuts to keep the height of the tallest ever bamboo as low as possible. The contribution in this paper is twofold, and is both theoretical and experimental. In particular, the focus is on understanding what has been called priority schedulings, i.e., cutting strategies where priority is given to bamboos whose current height is above a threshold greater than or equal to H = ∑ i = 1 n h i . Value H represents the total daily growth of the system and it is known that one cannot keep bamboos in the garden below this threshold indefinitely. As the first result, it is proved that, for any distribution of integer growth rates h 1 , ⋯ , h n and any priority scheduling, the system stabilises in a fixed cycle of cuts. Then, the focus is on the so-called ReduceMax strategy, a greedy priority scheduling that each day cuts the tallest bamboo, regardless of the growth rates distribution. ReduceMax is known to provide a O ( log n ) -approximation, with respect to the lower bound H. One of the main results achieved is that, if ReduceMax stabilises in a round-robin type cycle, then it guarantees 2-approximation. Furthermore, preliminary results are provided relating the structure of the input instance, in terms of growth rates, and the behavior of ReduceMax when applied to such inputs. Finally, a conjecture that ReduceMax is 2-approximating for the BGT problem is claimed, hence an extended experimental evaluation was conducted to support the conjecture and to compare ReduceMax with other relevant scheduling algorithms. The obtained results show that ReduceMax : (i) provides 2-approximation in all considered inputs; and (ii) always outperforms other considered strategies, even those for which better worst case approximation guarantees have been proven.

Bamboo Trimming Revisited: Simple Algorithms Can Do Well Too

“…This is the first time that a tight analysis has been achieved for Reduce-Fastest(x) for any value of x. For x = 2, the result gives a backlog of 3, which resolves a conjecture of [15]. On the other hand, we disprove the conjecture of [15] that Reduce-Fastest(1) achieves backlog 2.…”

“…Although there is a long history of studying resource-augmented variants of the cup game [9,13,14,26], it is only relatively recently that researchers have begun to study the resource-augmented fixed-rate version of the game [11,15,18]. These papers have dubbed of the problem as the Bamboo Garden Trimming Problem, based on the following (rather creative) problem interpretation.…”

“…x 2 4(x−1/2) ) for all x > 1, which yields a bound of 19/6 at x = 2 [11] (and is ≥ 1.25x for all x > 1). Extensive computer experimentation [15] suggests that this bound of 19/6 is still not optimal, and has led researchers to conjecture that Reduce-Fastest(2) actually achieves a backlog of 3. Based on the same experiments, the authors further conjecture that Reduce-Fastest(1) achieves the optimal backlog of 2 [15].…”

“…Extensive computer experimentation [15] suggests that this bound of 19/6 is still not optimal, and has led researchers to conjecture that Reduce-Fastest(2) actually achieves a backlog of 3. Based on the same experiments, the authors further conjecture that Reduce-Fastest(1) achieves the optimal backlog of 2 [15]. However, as of now, no theoretical bounds on the backlog of Reduce-Fastest(1) are known.…”

Special Issue on “Algorithm Engineering: Towards Practically Efficient Solutions to Combinatorial Problems”

“…Paper [3] deals with the bamboo garden trimming (BGT) problem, a periodic scheduling problem with several practical applications, e.g., in machine maintenance or cloud computing. The problem is known to be NP-hard due to its close relationship to another well-known optimization problem, namely, the pinwheel scheduling problem.…”