An Optimal Density Bound for Discretized Point Patrolling

TL;DR

This work resolves the discretized point patrolling problem in the covering setting by proving an optimal density bound of $D \ge \sum_{i=0}^{\infty} \frac{1}{2^i+1}$ (approximately $1.264$), improving the previous bound of $1.546$ and matching lower-bound constructions. It achieves this through novel folding techniques, including the cfold and an improved cfold^imp, together with careful density-loss analyses and computer-aided verification to reduce instances to finite checks. The paper also advances Bamboo Garden Trimming by giving a state-of-the-art $9/7$-approximation, built on a folding-based reduction to pinwheel packing and validated via precomputed schedules and obstruction lemmas. Collectively, these results settle the optimal density threshold for discretized point patrolling and push forward instance-dependent improvements in BGT, with implications for related cup-game and scheduling problems.

Abstract

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 $4/3$.

Theorems & Definitions (43)

• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• Lemma 2.1
• Proof 1
• Lemma 2.2
• Proof 2
• Lemma 3.1
• Proof 3
• Lemma 3.2