P-time event graphs (P-TEGs) are specific timed discrete-event systems, in which the timing of events is constrained by intervals. An important problem is to check, for all natural numbers d, the existence of consistent d-periodic trajectories for a given P-TEG. Let us consider a different, seemingly unrelated problem in graph theory: given three arbitrary square matrices P, I and C with elements in \mathbb{R\cup\{-\infty\}, find all real values of parameter \lambda such that the parametric directed graph having arcs weights of the form w((j,i)) = \max(P_{ij} + \lambda, I_{ij} - \lambda, C_{ij}) (for all arcs (j,i)) }  does not contain circuits with positive weight. In a related paper, we have proposed a strongly polynomial algorithm that solves the latter problem faster than other algorithms reported in literature. In the present paper, we show that the first problem can be formulated as an instance of the second; consequently, we prove that the same algorithm can be used to find d-periodic trajectories in P-TEGs faster than with previous approaches .
P-time event graphs (P-TEGs) are specific timed discrete-event systems, in which the timing of events is constrained by intervals. An important problem is to check, for all natural numbers d, the existence of consistent d-periodic trajectories for a given P-TEG. \cup\{-\infty\}, find all real values of parameter \lambda such that the parametric directed graph having arcs weights of the form w((j,i)) = \max(P_{ij} + \lambda, I_{ij} - \lambda, C_{ij}) (for all arcs (j,i)) } In graph theory, the Proportional-Inverse-Constant-Non-positive Circuit weight Problem (PIC-NCP) consists in finding all the values of a parameter such that a particular parametric weighted directed graph does not contain circuits with positive weight. In a related paper, we have proposed a strongly polynomial algorithm that solves the PIC-NCP in lower worst-case complexity compared to other algorithms reported in literature. In the present paper, we show that the first problem can be formulated as an instance of the second; consequently, we prove that the same algorithm can be used to find d-periodic trajectories in P-TEGs . Morever, exploiting the connection between the PIC-NCP and max-plus algebra, we prove that the existence of a consistent 1-periodic trajectory of a certain period is a necessary and sufficient condition for the existence of a consistent d-periodic trajectory of the same period, for any value of d .
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Let us consider a different, seemingly unrelated problem in graph theory: given three arbitrary square matrices P, I and C with elements in \mathbb{R
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In graph theory, the Proportional-Inverse-Constant-Non-positive Circuit weight Problem (PIC-NCP) consists in finding all the values of a parameter such that a particular parametric weighted directed graph
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latter problem faster than
PIC-NCP in lower worst-case complexity compared to
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faster than with previous approaches
. Morever, exploiting the connection between the PIC-NCP and max-plus algebra, we prove that the existence of a consistent 1-periodic trajectory of a certain period is a necessary and sufficient condition for the existence of a consistent d-periodic trajectory of the same period, for any value of d