NP-Hardness and a PTAS for the Pinwheel Problem

2026-04-15

AI Summary

This study addresses the computational complexity and approximation algorithms for the Pinwheel scheduling problem. By constructing a polynomial-time reduction from a known NP-hard problem, it rigorously establishes for the first time that Pinwheel scheduling is NP-hard, thereby implying the NP-hardness of several related periodic scheduling problems. Building on this hardness result, the paper presents a polynomial-time approximation scheme (PTAS) that surpasses the previously best-known approximation ratio of 9/7, achieving the current state-of-the-art approximation guarantee. This work not only settles the theoretical complexity status of the Pinwheel problem but also provides an efficient approximation framework for practical periodic task scheduling applications.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

In the pinwheel problem, one is given an $m$-tuple of positive integers $(a_1, \ldots, a_m)$ and asked whether the integers can be partitioned into $m$ color classes $C_1,\ldots,C_m$ such that every interval of length $a_i$ has non-empty intersection with $C_i$, for $i = 1, 2, \ldots, m$. It was a long-standing open question whether the pinwheel problem is NP-hard. We affirm a prediction of Holte et al. (1989) by demonstrating, for the first time, NP-hardness of the pinwheel problem. This enables us to prove NP-hardness for a host of other problems considered in the literature: pinwheel covering, bamboo garden trimming, windows scheduling, recurrent scheduling, and the constant gap problem. On the positive side, we develop a PTAS for an approximate version of the pinwheel problem. Previously, the best approximation factor known to be achievable in polynomial time was $\frac{9}{7}$.

Problem

• Pinwheel Problem
• NP-Hardness
• Scheduling
• Computational Complexity
• Approximation Algorithms

Innovation

• NP-hardness
• PTAS
• Pinwheel Problem
• Approximation Algorithm
• Scheduling