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

TL;DR

This paper presents a max-plus algebra based algorithm to efficiently solve the Non-positive Circuit weight Problem in parametric graphs, with applications in discrete-event system analysis, achieving a strongly polynomial time complexity.

Contribution

It introduces a novel algorithm that generalizes previous NCP solutions for parametric graphs, improving complexity to strongly polynomial time using dioid theory.

Findings

Algorithm solves NCP for a broad class of parametric graphs.

Achieves strongly polynomial time complexity of O(n^4).

Applicable to analysis of P-time event graphs.

Abstract

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 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).