Combinatorial Perpetual Scheduling

TL;DR

The paper develops a combinatorial framework for perpetual scheduling (CBGT) and its pinwheel variant (CPS) by tying growth rates to matroid polytopes. It proves a tight height bound of $h( obreak pi)<2$ for all matroid CBGT instances, with efficient poly-time algorithms achieving height $2$ for uniform/partition and height $4$ for graphic/laminar matroids. For general set systems, the authors show an optimal $Θ( obreak ext{log}|E|)$ height bound with a constructive, oracle-based algorithm, and they connect these results to density thresholds in CPS. A novel discrepancy-matroid construction and a matroid-intersection framework underpin the existential result, while rainbow-circuit-free colorings and Fuse–Unfuse scheduling yield practical poly-time implementations for key matroid classes. Overall, the work clarifies the trade-offs between combinatorial structure and schedulability, and it furnishes near-optimal strategies for a broad spectrum of perpetual-scheduling problems with applications to density bounds in CPS.

Abstract

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.

Figures (3)

• Figure 1: $G_e$ for $g(e) = 4/11$ and $T=22$
• Figure 2: Binary trees obtained in the preprocessing phase of \ref{['alg:fun']} for the 2-uniform matroid with ground set $\{a,b,c,d,e\}$.
• Figure 3: Rainbow-circuit-free coloring of a graphic matroid on edge set $\{e_1, e_2, \dots, e_{10}\}$.

Theorems & Definitions (76)

• Theorem 1.1
• Theorem 1.1: $\star$
• Theorem 1.1
• Proposition 2.0
• proof
• Corollary 2.0: $\star$
• Corollary 2.0: $\star$
• Theorem 3.1
• Lemma 3.1
• proof