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

Authors Mirabel Mendoza-Cadena , Arturo Merino , Mads Anker Nielsen , Kevin Schewior

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ACM Subject Classification
  • Theory of computation → Scheduling algorithms
Keywords
  • Perpetual Scheduling
  • Matroids
  • Bamboo Garden Trimming
  • Pinwheel

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Abstract

This paper considers a framework for combinatorial variants of perpetual-scheduling problems. Given an independence system (E,ℐ), a schedule consists of an independent set I_t ∈ ℐ for every time step t ∈ 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 ∈ 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 ℐ.
Using the integrality of the matroid-intersection polytope, we prove that, when (E,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 Θ(log |E|) and can be achieved by an efficient algorithm. For combinatorial pinwheel scheduling, where each element e ∈ E needs to occur in the schedule at least every a_e ∈ ℕ time steps, our results imply bounds on the density sufficient for schedulability.

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Mirabel Mendoza-Cadena, Arturo Merino, Mads Anker Nielsen, and Kevin Schewior. Combinatorial Perpetual Scheduling: Existence and Computation of Low-Height Schedules. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 142:1-142:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.142

Author Details

Mirabel Mendoza-Cadena
  • Instituto de Ciencias de la Ingeniería, Universidad de O'Higgins, Rancagua, Chile
Arturo Merino
  • Department of Computer Science, University of Chile, Santiago, Chile
Mads Anker Nielsen
  • Department of Mathematics and Computer Science, University of Cologne, Germany
Kevin Schewior
  • Department of Mathematics and Computer Science, University of Cologne, Germany
  • Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark

Funding

  • Mendoza-Cadena, Mirabel: Centro de Modelamiento Matemático (CMM) BASAL fund FB210005 for center of excellence from ANID-Chile.
  • Merino, Arturo: Supported by ANID FONDECYT Iniciación No. 11251528.
  • Schewior, Kevin: Supported in part by the Independent Research Fund Denmark, Natural Sciences, grant DFF-4283-00079B.

Acknowledgements

We thank Tobias Mömke, Victor Verdugo, and José Verschae for organizing the Santiago Summer Workshop on Combinatorial Optimization 2025, where this research began. We especially thank Sofía Errázuriz for valuable discussions.

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