参数图中的非正回路权重问题：基于dioid理论的解决方案
The non-positive circuit weight problem in parametric graphs: a solution based on dioid theory
AI总结：
本文提出基于dioid理论的算法，用于解决参数图中的非正回路权重问题，实现强多项式时间复杂度。
摘要：
让我们考虑一个参数加权有向图，其中每条边$(j,i)$的权重形式为$w((j,i))=\max(P_{ij}+λ,I_{ij}-λ,C_{ij})$，其中$λ$是一个实参数，$P$、$I$和$C$是任意平方矩阵，元素属于$\mathbb{R}\cup\{-\infty\}$。本文设计了一个算法，用于解决此类参数图中的非正回路权重问题（NCP），即找出所有$λ$值，使得图中不包含权重为正的回路。该问题扩展了文献中已研究的其他NCP实例，应用于分析称为P-time事件图的一类离散事件系统的一致性。所提出的算法基于最大-加法代数和形式语言，并改进了其他现有方法的最坏情况复杂度，达到强多项式时间复杂度$\mathcal{O}(n^4)$（其中$n$是图中的节点数）。
英文摘要：
Let us consider a parametric weighted directed graph in which every arc $(j,i)$ has weight of the form $w((j,i))=\max(P_{ij}+λ,I_{ij}-λ,C_{ij})$, where $λ$ 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 $λ$ 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 improves the worst-case complexity of 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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