Combinatorial Perpetual Scheduling

2026-02-12

AI Summary

This work addresses the perpetual scheduling problem under combinatorial constraints: at each time step, an independent set from a set system $(E, \mathcal{I})$ must be selected to satisfy prescribed frequency requirements for all elements while minimizing the maximum cumulative load (height) over any element. The study generalizes perpetual scheduling to arbitrary combinatorial constraints by leveraging the integrality of matroid intersection polyhedra. It achieves optimal schedules with maximum height 2 for uniform and partition matroids, and height 4 for graphic and laminar matroids. For general set systems, the paper establishes a tight $\Theta(\log|E|)$ upper bound on the achievable maximum height and derives schedulability density bounds for combinatorial round-robin scheduling.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $\Theta(\log |E|)$ 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.

Problem

• combinatorial perpetual scheduling
• bamboo garden trimming
• matroid
• pinwheel scheduling
• maximum height minimization

Innovation

• combinatorial perpetual scheduling
• bamboo garden trimming
• matroid intersection
• integrality gap
• pinwheel scheduling