Periodic trajectories in P-time event graphs and the non-positive circuit weight problem

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.