Finite Pinwheel Scheduling: the k-Visits Problem

TL;DR

The paper studies deadline-based scheduling by introducing the finite variant $k$-Visits, focusing on the challenging case $k=2$ (2-Visits). It proves that 2-Visits is strongly NP-complete via a chain of reductions from Restricted Numerical 3-Dimensional Matching (RN3DM) to IN3DM to Position Matching and finally to 2-Visits, and extends these hardness results to generalized versions where deadlines may vary across the schedule. It also identifies tractable regimes: linear-time solvability for distinct deadlines, an FPT algorithm parameterized by the maximum cluster size of the discretized deadline sequence, and linear-time solutions when there are at most two distinct deadlines. Additionally, it shows that a linear-time reduction to Position Matching yields efficient algorithms for special cases and enables hardness results to transfer to generalized Pinwheel Scheduling variants, including Var-$k$-Visits and Threshold Pinwheel Scheduling. Collectively, these results shed light on the boundary between tractable and intractable deadline-based scheduling and offer new tools towards understanding the classic Pinwheel Scheduling problem.

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 $k\geq 2$ 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.

Figures (3)

• Figure 1: A map of our reductions. $\textnormal{RN3DM}$ (Def. \ref{['def:rn3dm']}) and $\textnormal{IN3DM}$ (Def. \ref{['def:numerical_ineq']}) are variants of $\textnormal{Numerical\ 3-Dimensional Matching}$. Reductions denoted by red arrows are used to prove strong NP-completeness for $\textnormal{2-Visits}$ (Section \ref{['sec:two_visits_hardness']}). The blue-arrow reduction is a linear time Turing reduction from $\textnormal{2-Visits}$ to $\textnormal{Position Matching}$, and is the backbone of the algorithms we propose for $2$-Visits (Section \ref{['sec:two_visits_algo']}). Lastly, the green-arrow reductions prove strong NP-hardness for generalizations of $k$-Visits and Pinwheel Scheduling (Section \ref{['sec:hardness_extra']}).
• Figure 2: A $\textnormal{2-Visits}$ instance $D$, with its discretized sequence $A$ consisting of three clusters (marked with red). $T$ is the respective set of targets (gaps). Green lines among $D,A,T$ show triplets that constitute solutions to each of the three Position Matching instances obtained through Theorem \ref{['theorem:reduction']}.
• Figure 3: The sequence $A$ of Fig. \ref{['fig:matching']} with its gaps and the schedule $S$ found by solving the Position Matching instances. Primary visits are denoted with orange and secondary visits with blue. For simplicity, we use the nodes' deadlines to show the schedule (instead of their indices).

Theorems & Definitions (69)

• Definition 1: $\textnormal{Pinwheel Scheduling}$ Holte_Pinwheel
• Definition 2: Density
• Definition 3: $\textnormal{N3DM}$
• Definition 4: $\textnormal{$k$-Visits}$
• Corollary 1
• Definition 5: Discretized Sequence
• Definition 6: $\textnormal{2-Visits}$
• Lemma 1
• proof
• Lemma 2