Finite Pinwheel Covering

2026-07-30

AI Summary

This study addresses a finite variant of the Pinwheel Covering problem, where each task with frequency \( f_i \) must appear at most once in any consecutive \( f_i \) time slots and is required to be scheduled exactly \( k \) times. The authors introduce the \( k \)-Visits Covering model and establish its strong NP-completeness even for \( k = 2 \), highlighting a fundamental distinction from classical bin-packing problems. They devise linear-time algorithms for two special classes of frequency structures and present a randomized polynomial-time algorithm under constant frequencies. Furthermore, they disprove the existence of any nontrivial density upper bound, thereby refuting a natural conjecture—inspired by the classic Pinwheel problem—that a density threshold governs feasibility in this finite setting.

Abstract

In perpetual scheduling theory, the Pinwheel Covering problem asks, given $n$ frequencies $f_i$, whether there exists an infinite schedule such that every $f_i$ consecutive entries contain at most one occurrence of $i\in [n]$. This models $n$ agents taking turns at executing a job, with a recovery period before working again. Pinwheel Covering is, in a sense, the dual of Pinwheel Packing (also known as Pinwheel Scheduling), which similarly asks for at least one occurrence of $i$ in every $f_i$ consecutive entries. The complexity of both problems is a major open question: both are known to be in PSPACE, but PSPACE-hardness remains unknown. Recently, a finite version of Pinwheel Packing requiring only $k$ occurrences of $i\in [n]$ was introduced by [Kanellopoulos et al., SODA 2026] and proven to be strongly NP-complete. In this work we introduce $k$-Visits Covering, the analogous finite version of Pinwheel Covering, establishing strong NP-completeness even for $k=2$. As a corollary, we obtain that a generalization of Pinwheel Covering with varying frequencies is strongly NP-hard. To the best of our knowledge, this is the first strong NP-hardness result in the covering setting. We complement these results with a linear-time algorithm for $2$-Visits Covering with two distinct frequencies and a randomized polynomial-time algorithm when the number of distinct frequencies is constant. Lastly, we study the density thresholds of $k$-Visits Covering and prove that no non-trivial density bounds exist, contrasting the finite packing version.

Problem

• Pinwheel Covering
• k-Visits Covering
• strong NP-completeness
• finite scheduling
• computational complexity

Innovation

• k-Visits Covering
• strong NP-completeness
• finite scheduling
• density threshold
• perpetual scheduling