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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 areal parameter and $P$, $I$ and $C$ are arbitrary square matrices with elementsin $\mathbb{R}\cup\{-\infty\}$. In this paper, we design an algorithm thatsolves the Non-positive Circuit weight Problem (NCP) on this class ofparametric graphs, which consists in finding all values of $\lambda$ such thatthe graph does not contain circuits with positive weight. This problem, whichgeneralizes other instances of the NCP previously investigated in theliterature, has applications in the consistency analysis of a class ofdiscrete-event systems called P-time event graphs. The proposed algorithm isbased on max-plus algebra and formal languages, and improves the worst-casecomplexity of other existing approaches, achieving strongly polynomial timecomplexity $\mathcal{O}(n^4)$ (where $n$ is the number of nodes in the graph).

The non-positive circuit weight problem in parametric graphs: a solution based on dioid theory