NP-Hardness and a PTAS for the Pinwheel Problem

TL;DR

This paper proves the NP-hardness of the long-standing open problem called the pinwheel problem and introduces a polynomial-time approximation scheme (PTAS) for its approximate version.

Contribution

It establishes NP-hardness for the pinwheel problem and related problems, and develops a PTAS improving previous approximation factors.

Findings

NP-hardness of the pinwheel problem is demonstrated.

A PTAS is developed for an approximate version of the problem.

Previous best polynomial-time approximation factor was 9/7.

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 $\frac{9}{7}$.