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 $\frac{9}{7}$.

翻译：在风车问题中，给定一个$m$元正整数元组$(a_1, \ldots, a_m)$，询问是否可将这些整数划分为$m$个颜色类$C_1,\ldots,C_m$，使得对于每个$i = 1, 2, \ldots, m$，任意长度为$a_i$的区间都与$C_i$有非空交集。风车问题是否为NP-困难这一长期悬而未决的问题，我们首次通过证明其NP-困难性，验证了Holte等人（1989）的预测。由此可推导出文献中诸多其他问题的NP-困难性：风车覆盖问题、竹园修剪问题、窗口调度问题、循环调度问题以及常间隙问题。在积极方面，我们为风车问题的近似版本设计了一个多项式时间近似方案（PTAS）。此前，多项式时间内可达到的最佳近似因子为$\frac{9}{7}$。