Hardness, Tractability and Density Thresholds of finite Pinwheel Scheduling Variants

2026-04-17

AI Summary

This study addresses the k-Visits Pinwheel scheduling problem, which asks whether there exists a schedule of length kn in which each of n tasks is executed exactly k times without violating its deadline. Leveraging computational complexity theory, parameterized algorithms, and probabilistic methods, the authors establish that 2-Visits is strongly NP-complete when the maximum multiplicity is two. They propose a randomized polynomial-time (RP) algorithm for instances with a constant number of distinct deadlines and extend their analysis to a generalized model with heterogeneous visit requirements. Furthermore, they derive bounds on the critical density threshold: for 2-Visits, the lower bound is approximately 0.9142, and as k tends to infinity, the threshold for k-Visits converges to 5/6.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

The k-Visits problem is a recently introduced finite version of Pinwheel Scheduling [Kanellopoulos et al., SODA 2026]. Given the deadlines of n tasks, the problem asks whether there exists a schedule of length kn executing each task exactly k times, with no deadline expiring between consecutive visits (executions) of each task. In this work we prove that 2-Visits is strongly NP-complete even when the maximum multiplicity of the input is equal to 2, settling an open question from [Kanellopoulos et al., SODA 2026] and contrasting the tractability of 2-Visits for simple sets. On the other hand, we prove that 2-Visits is in RP when the number of distinct deadlines is constant, thus making progress on another open question regarding the parameterization of 2-Visits by the number of numbers. We then generalize all existing positive results for 2-Visits to a version of the problem where some tasks must be visited once and some other tasks twice, while providing evidence that some of these results are unlikely to transfer to 3-Visits. Lastly, we establish bounds for the density thresholds of k-Visits, analogous to the $(5/6)$-threshold of Pinwheel Scheduling [Kawamura, STOC 2024]; in particular, we show a $\sqrt{2}-1/2\approx 0.9142$ lower bound for the density threshold of 2-Visits and prove that the density threshold of k-Visits approaches $5/6\approx 0.8333$ for $k \to \infty$.

Problem

• Pinwheel Scheduling
• k-Visits
• NP-completeness
• density threshold
• tractability

Innovation

• k-Visits
• NP-completeness
• density threshold
• parameterized complexity
• Pinwheel Scheduling