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

TL;DR

This work generalizes Gallai's finite-constraint feasibility criterion to arbitrary infinite precedence graphs, demonstrating that a real solution to systems of precedence inequalities exists if and only if the associated precedence graph contains no $\infty$-weight paths. It then specializes to two practically relevant infinite-graph classes: $\mathbb{N}$-periodic and ultimately periodic graphs, proving that the existence of solutions can be verified in strongly polynomial time $O(n^5)$ by computing matrix sequences $\Pi(h)$ and related constructs. These results yield efficient, scalable methods for testing the consistency of P-time event graphs (P-TEGs) under both loose and strict initial conditions, with direct implications for modeling cyclic production systems under time windows. The paper also highlights connections to vector addition systems with states and suggests avenues for extending the approach to broader dynamic-graph models and scheduling problems.

Abstract

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.

Figures (11)

• Figure 1: Infinite precedence graph with an $\infty$-weight path but without circuits with positive weight.
• Figure 2: Application of function $\Phi$ illustrated on a finite precedence graph. Whenever not indicated, the weight of arcs is $0$. Blue thick arcs highlight the difference between $\mathcal{G}(A)$ and $\mathcal{G}(A^*)$, and between $\mathcal{G}(A^*)$ and $\mathcal{G}(\Phi(A))$.
• Figure 3: Schematic representation of $\mathcal{G}(A^*)$ and $\mathcal{G}(\Phi(A))$ for an example of graph $\mathcal{G}(A^*)$ consisting of three maximal strongly connected subgraphs $\mathcal{G}_1,\mathcal{G}_2,\mathcal{G}_3$. Compared to $\mathcal{G}(A^*)$, $\mathcal{G}(\Phi(A))$ has an additional arc $(j,i)$ of weight $-(A^*)_{ji}$, where $i$ and $j$ belong to $\mathcal{G}_2$ and $\mathcal{G}_3$, respectively. Observe that $\mathcal{G}(\Phi(A))$ has only two maximal strongly connected subgraphs.
• Figure 4: Precedence graph corresponding to the difference bound matrix of \ref{['ex:potential_inequalities_1']}.
• Figure 5: Static graph and corresponding $\mathbb{N}$- and $\mathbb{Z}$-periodic graphs. Every arc $e$ in the static graph is labeled "$\textup{shift}(e),w(e)$". Every node $v$ in the periodic graphs is labeled "$(\hbox{base}(v),\textup{shift}(v))$".

Theorems & Definitions (34)

• remark 1
• theorem 1
• remark 2
• theorem 2
• definition 1
• lemma 1
• proof
• definition 2
• lemma 2
• proof