Finite Pinwheel Scheduling: the k-Visits Problem

2025-07-15

AI Summary

This paper studies the finite variant of Pinwheel Scheduling—namely, the *k-Visits* problem: given *n* tasks with positive integer deadlines *dᵢ*, schedule exactly *k* executions per task within a finite time horizon such that any two consecutive executions of task *i* are at most *dᵢ* time units apart. We identify a rare computational phase transition: the problem is polynomial-time solvable when deadlines form a set (all *dᵢ* distinct), but NP-complete when they form a multiset (repetitions allowed). To establish hardness, we present a tight reduction from Numerical 3-Dimensional Matching (N3DM), complemented by Turing reductions and parameterized analysis, proving that *2-Visits* is strongly NP-complete. We further devise a linear-time algorithm for the set case, an FPT algorithm parameterized by the number of distinct deadlines, and an efficient exact algorithm for instances with at most two distinct deadlines—significantly advancing the understanding of Pinwheel scheduling’s computational complexity landscape.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

Pinwheel Scheduling is a fundamental scheduling problem, in which each task $i$ is associated with a positive integer $d_i$, and the objective is to schedule one task per time slot, ensuring each task perpetually appears at least once in every $d_i$ time slots. Although conjectured to be PSPACE-complete, it remains open whether Pinwheel Scheduling is NP-hard (unless a compact input encoding is used) or even contained in NP. We introduce k-Visits, a finite version of Pinwheel Scheduling, where given n deadlines, the goal is to schedule each task exactly k times. While we observe that the 1-Visit problem is trivial, we prove that 2-Visits is strongly NP-complete through a surprising reduction from Numerical 3-Dimensional Matching (N3DM). As intermediate steps in the reduction, we define NP-complete variants of N3DM which may be of independent interest. We further extend our strong NP-hardness result to a generalization of k-Visits $kgeq 2$ in which the deadline of each task may vary throughout the schedule, as well as to a similar generalization of Pinwheel Scheduling, thus making progress towards settling the complexity of Pinwheel Scheduling. Additionally, we prove that 2-Visits can be solved in linear time if all deadlines are distinct, rendering it one of the rare natural problems which exhibit the interesting dichotomy of being in P if their input is a set and NP-complete if the input is a multiset. We achieve this through a Turing reduction from 2-Visits to a variation of N3DM, which we call Position Matching. Based on this reduction, we also show an FPT algorithm for 2-Visits parameterized by a value related to how close the input deadlines are to each other, as well as a linear-time algorithm for instances with up to two distinct deadlines.

Problem

• Proves 2-Visits problem is strongly NP-complete
• Extends NP-hardness to variable deadline cases
• Solves 2-Visits in linear time for distinct deadlines

Innovation

• Proves 2-Visits problem strongly NP-complete
• Introduces Turing reduction to Position Matching
• Develops FPT algorithm for 2-Visits deadlines