Let us consider a parametric weighted directed graph in which every arc $(j,i)$ has weight of the form $w((j,i))=\max(P_{ij}+\lambda,I_{ij}-\lambda,C_{ij})$, where $\lambda$ is a real parameter and $P$, $I$ and $C$ are arbitrary square matrices with elements in $\mathbb{R}\cup\{-\infty\}$. In this paper, we design an algorithm that solves the Non-positive Circuit weight Problem (NCP) on this class of parametric graphs, which consists in finding all values of $\lambda$ such that the graph does not contain circuits with positive weight. This problem, which generalizes other instances of the NCP previously investigated in the literature, has applications in the consistency analysis of a class of discrete-event systems called P-time event graphs. The proposed algorithm is based on max-plus algebra and formal languages and runs faster than other existing approaches, achieving strongly polynomial time complexity $\mathcal{O}(n^4)$ (where $n$ is the number of nodes in the graph).

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参数图中的非正弦电路权重问题：基于二分体理论的快速解决方案

让我们考虑一个参数加权有向图，其中每个弧$（j，i）$的权重形式为$ w（（j，i））= \ max（P_ {ij} + \ lambda，I_ {ij}- \ lambda，C_ {ij}）$，其中$ \ lambda $是实参，$ P $，$ I $和$ C $是任意平方矩阵，元素在$ \ mathbb {R} \ cup \ {-\ infty \} $。在本文中，我们设计了一种算法来解决此类参数图上的非正电路权重问题（NCP），该算法包括找到$ \ lambda $的所有值，以使该图不包含权重为正的电路。这个问题概括了先前在文献中研究过的NCP的其他实例，已在一类称为P时间事件图的离散事件系统的一致性分析中得到了应用。