This paper introduces a framework for combinatorial variants of perpetual-scheduling problems. Given a set 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 $e \in E$ given as a convex combination of incidence vectors of $\mathcal{I}$ and are reset to zero when scheduled, with the goal of minimizing the maximum height attained by any element. 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 set systems, the optimal guaranteed height is $Θ(\log |E|)$ 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.

翻译：本文提出了一种用于持续调度问题的组合变体框架。给定集合系统$(E,\mathcal{I})$，调度由每个时间步$t \in \mathbb{N}$中的独立集$I_t \in \mathcal{I}$构成，其目标是满足$E$中元素出现频率的要求。我们特别关注组合式竹林修剪问题，其中元素以生长速率$g(e)$（$e \in E$）累积高度，该速率由$\mathcal{I}$的关联向量的凸组合给出，并在被调度时重置为零，目标是最小化任意元素达到的最大高度。利用拟阵交多面体的整性，我们证明当$(E,\mathcal{I})$为拟阵时，可以保证至多2的最大高度，且该界是最优的。我们通过针对特定拟阵类的高效算法对这一存在性结果进行补充：在均匀拟阵和划分拟阵中实现最大高度2，在图形拟阵和层状拟阵中实现最大高度4。相比之下，我们证明对于一般集合系统，最优保证高度为$Θ(\log |E|)$，且可通过高效算法实现。对于组合式风车调度问题（其中每个元素$e\in E$需要至少每$a_e \in \mathbb{N}$个时间步出现在调度中），我们的结果推导出了可调度性所需的密度界限。