The pinwheel problem is a real-time scheduling problem that asks, given $n$ tasks with periods $a_i \in \mathbb{N}$, whether it is possible to infinitely schedule the tasks, one per time unit, such that every task $i$ is scheduled in every interval of $a_i$ units. We study a corresponding version of this packing problem in the covering setting, stylized as the discretized point patrolling problem in the literature. Specifically, given $n$ tasks with periods $a_i$, the problem asks whether it is possible to assign each day to a task such that every task $i$ is scheduled at \textit{most} once every $a_i$ days. The density of an instance in either case is defined as the sum of the inverses of task periods. Recently, the long-standing $5/6$ density bound conjecture in the packing setting was resolved affirmatively. The resolution means any instance with density at least $5/6$ is schedulable. A corresponding conjecture was made in the covering setting and renewed multiple times in more recent work. We resolve this conjecture affirmatively by proving that every discretized point patrolling instance with density at least $\sum_{i = 0}^{\infty} 1/(2^i + 1) \approx 1.264$ is schedulable. This significantly improves upon the current best-known density bound of 1.546 and is, in fact, optimal. We also study the bamboo garden trimming problem, an optimization variant of the pinwheel problem. Specifically, given $n$ growth rates with values $h_i \in \mathbb{N}$, the objective is to minimize the maximum height of a bamboo garden with the corresponding growth rates, where we are allowed to trim one bamboo tree to height zero per time step. We achieve an efficient $9/7$-approximation algorithm for this problem, improving on the current best known approximation factor of $4/3$.

翻译：风车问题是一个实时调度问题，它询问给定n个周期为a_i∈ℕ的任务，是否能够无限地调度这些任务（每个时间单位调度一个），使得每个任务i在每个长度为a_i的时间区间内都被调度。我们研究该装箱问题在覆盖设置下的对应版本，在文献中被形式化为离散化点巡逻问题。具体而言，给定n个周期为a_i的任务，该问题询问是否能够为每一天分配一个任务，使得每个任务i至多每a_i天被调度一次。在这两种情况下，实例的密度被定义为任务周期倒数的和。最近，装箱设置中长期存在的5/6密度界猜想被肯定地解决。该解决意味着任何密度至少为5/6的实例都是可调度的。在覆盖设置中提出了相应的猜想，并在近期工作中多次被重新讨论。我们通过证明每个密度至少为∑_{i=0}^{∞} 1/(2^i+1)≈1.264的离散化点巡逻实例都是可调度的，从而肯定地解决了这一猜想。这显著改进了当前已知的最佳密度界1.546，并且实际上是最优的。我们还研究了竹子园修剪问题，这是风车问题的一个优化变体。具体而言，给定n个生长速率h_i∈ℕ，目标是最小化具有相应生长速率的竹子园的最大高度，其中我们被允许每个时间步将一棵竹子修剪至零高度。我们为该问题设计了一个高效的9/7近似算法，改进了当前已知的4/3近似因子。