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.

中文翻译：

---

P时间事件图中的周期性轨迹和非正电路权重问题

P时间事件图（P-TEG）是特定的定时离散事件系统，其中事件的时间受到间隔的限制。一个重要的问题是对于给定的P-TEG，检查所有自然数$ d $是否存在一致的$ d $周期轨迹。让我们考虑图论中的一个不同的，看似无关的问题：给定三个任意平方矩阵$ P $，$ I $和$ C $，且元素在$ \ mathbb {R} \ cup \ {-\ infty \} $中，找到参数$ \ lambda $的所有实数值，使得具有弧线权重的参数有向图的形式为$ w（（j，i））= \ max（P_ {ij} + \ lambda，I_ {ij}-\ lambda， C_ {ij}）$（对于所有弧$（j，i）$）不包含权重为正的电路。在相关的论文中，我们提出了一种强多项式算法，该算法比文献中报道的其他算法更快地解决了后一个问题。在本文中，我们表明可以将第一个问题表达为第二个问题的实例。因此，我们证明了使用相同的算法可以比以前的方法更快地找到P-TEG中的$ d $周期轨迹。