NP-Hardness and a PTAS for the Pinwheel Problem

Authors: Robert Kleinberg, Ahan Mishra

DOI Link

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 \(\frac{9}{7}\).

Keywords

• approximation algorithms
• hardness
• theoretical computer science
• pinwheel problem

Recommended citation: Robert Kleinberg, and Ahan Mishra, "NP-Hardness and a PTAS for the Pinwheel Problem," 2026 IEEE Symposium on Foundations of Computer Science (FOCS), New York City, New York, 2026. http://ahanmishra.com/files/hardness-paper.pdf