Hardness, Tractability and Density Thresholds of finite Pinwheel Scheduling Variants

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 $k \to \infty$.