NP-Hardness and a PTAS for the Pinwheel Problem
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908966833881088 |
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| author | Kleinberg, Robert Mishra, Ahan |
| author_facet | Kleinberg, Robert Mishra, Ahan |
| contents | 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 $\frac{9}{7}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_13974 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | NP-Hardness and a PTAS for the Pinwheel Problem Kleinberg, Robert Mishra, Ahan Data Structures and Algorithms Computational Complexity 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 $\frac{9}{7}$. |
| title | NP-Hardness and a PTAS for the Pinwheel Problem |
| topic | Data Structures and Algorithms Computational Complexity |
| url | https://arxiv.org/abs/2604.13974 |