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<p>Precedence constraints are inequalities used to model time dependencies. In 1958, Gallai proved that a finite system of precedence constraints admits solutions if and only if the corresponding precedence graph does not contain positive-weight circuits. We show that this result extends naturally to the case of infinitely many constraints. We then analyze two specific classes of infinite precedence graphs – <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {N}\)</EquationSource> </InlineEquation>-periodic and ultimately periodic graphs – and prove that the existence of solutions of their related constraints can be verified in strongly polynomial time. The obtained algorithms find applications in P-time event graphs, which are a subclass of P-time Petri nets able to model production systems under cyclic schedules where tasks need to be performed within given time windows.</p>

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Infinite precedence graphs for consistency verification in P-time event graphs

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月23日

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  • Davide Zorzenon,
  • Jörg Raisch

摘要

Precedence constraints are inequalities used to model time dependencies. In 1958, Gallai proved that a finite system of precedence constraints admits solutions if and only if the corresponding precedence graph does not contain positive-weight circuits. We show that this result extends naturally to the case of infinitely many constraints. We then analyze two specific classes of infinite precedence graphs – \(\mathbb {N}\) -periodic and ultimately periodic graphs – and prove that the existence of solutions of their related constraints can be verified in strongly polynomial time. The obtained algorithms find applications in P-time event graphs, which are a subclass of P-time Petri nets able to model production systems under cyclic schedules where tasks need to be performed within given time windows.