Combinatorial Perpetual Scheduling: Existence and Computation of Low-Height Schedules

TL;DR

This paper studies combinatorial perpetual scheduling, proving optimal bounds for matroids and providing algorithms for various classes, with implications for pinwheel scheduling.

Contribution

It establishes existence and computation of low-height schedules in combinatorial perpetual scheduling, especially for matroids and specific independence systems.

Findings

Maximum height of 2 for matroids, which is optimal.

Efficient algorithms for uniform and partition matroids achieving height 2.

For general systems, the optimal height is Θ(log |E|).

Abstract

This paper considers a framework for combinatorial variants of perpetual-scheduling problems. Given an independence system $(E,\mathcal{I})$, a schedule consists of an independent set $I_t \in \mathcal{I}$ for every time step $t \in \mathbb{N}$, with the objective of fulfilling frequency requirements on the occurrence of elements in $E$. We focus specifically on combinatorial bamboo garden trimming, where elements accumulate height at growth rates $g(e)$ for $e \in E$ and are reset to zero when scheduled, with the goal of minimizing the maximum height attained by any element. We assume that $g$ is normalized so that it is a convex combination of the incidence vectors of $\mathcal{I}$. Using the integrality of the matroid-intersection polytope, we prove that, when $(E,\mathcal{I})$ is a matroid, it is possible to guarantee a maximum height of at most 2, which is optimal. We complement this existential result with efficient algorithms for specific matroid classes, achieving a maximum height of 2 for uniform and partition matroids, and 4 for graphic and laminar matroids. In contrast, we show that for general independence systems, the optimal guaranteed height is $\Theta(\log |E|)$ and can be achieved by an efficient algorithm. For combinatorial pinwheel scheduling, where each element $e\in E$ needs to occur in the schedule at least every $a_e \in \mathbb{N}$ time steps, our results imply bounds on the density sufficient for schedulability.