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arXiv · 2510.06533

A Computer-Assisted Proof of the Optimal Density Bound for Pinwheel Covering

Akitoshi Kawamura·Yusuke Kobayashi
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Abstract

In the covering version of the pinwheel scheduling problem, a daily task must be assigned to agents under the constraint that agent $i$ can perform the task at most once in any $a_i$-day interval. In this paper, we determine the optimal constant $\alpha^* = 1.264\ldots$ such that every instance with $\sum_{i} 1 / a_i \ge \alpha^*$ is schedulable. This resolves an open problem posed by Kawamura and Soejima (2020). Our proof combines Kawamura's (2026) techniques for the packing version with new mathematical insights to reduce the analysis to a finite set of instances, which are then verified through an exhaustive computer-aided search that draws on ideas from G\k{a}sieniec, Smith, and Wild (2022). The same result was obtained independently by Mishra (2026).

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Akitoshi Kawamura, Yusuke Kobayashi. 2025-10-08. A Computer-Assisted Proof of the Optimal Density Bound for Pinwheel Covering. https://arxiv.org/abs/2510.06533

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