Periodic trajectories in P-time event graphs and the non-positive circuit weight problem
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866912886047113216 |
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| author | Zorzenon, Davide Komenda, Jan Raisch, Joerg |
| author_facet | Zorzenon, Davide Komenda, Jan Raisch, Joerg |
| contents | 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. 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. Moreover, exploiting the connection between the PIC-NCP and max-plus algebra we prove that, given a P-TEG, 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 $d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_01024 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Periodic trajectories in P-time event graphs and the non-positive circuit weight problem Zorzenon, Davide Komenda, Jan Raisch, Joerg Data Structures and Algorithms Discrete Mathematics Systems and Control Optimization and Control 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. 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. Moreover, exploiting the connection between the PIC-NCP and max-plus algebra we prove that, given a P-TEG, 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 $d$. |
| title | Periodic trajectories in P-time event graphs and the non-positive circuit weight problem |
| topic | Data Structures and Algorithms Discrete Mathematics Systems and Control Optimization and Control |
| url | https://arxiv.org/abs/2103.01024 |