An Optimal Density Bound for Discretized Point Patrolling

2025-10-24

AI Summary

This paper investigates two periodic scheduling problems: Discrete Point Patrolling (DPP) and Bamboo Trimming. For DPP, we establish—via rigorous analysis—the tight lower bound of 1.264 on the density threshold for infinite schedulability, refuting the long-standing conjecture that 1.546 is an upper bound. For Bamboo Trimming, we propose a novel greedy algorithm that improves the best-known approximation ratio from 4/3 ≈ 1.333 to 9/7 ≈ 1.286. Methodologically, we integrate real-time schedulability analysis, density-bound theory, and combinatorial optimization to derive tight characterizations linking task density to schedulability feasibility. Collectively, these results advance the theoretical foundations of density-based schedulability criteria in periodic real-time systems and sharpen the known algorithmic limits for both problems.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 extit{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 $4/3$.

Problem

• Resolving the density bound conjecture for discretized point patrolling
• Proving schedulability for density above approximately 1.264
• Developing an improved approximation algorithm for bamboo trimming

Innovation

• Proved optimal density bound for discretized point patrolling
• Achieved efficient approximation algorithm for bamboo trimming
• Resolved long-standing conjecture in covering scheduling problem