Extremality criteria for the supereigenvector space in max-plus algebra

TL;DR

This paper establishes necessary and sufficient conditions for extremality of supereigenvectors in max-plus algebra and provides an efficient $O(n^2)$ verification method.

Contribution

It introduces extremality criteria for max-algebraic supereigenvectors and offers a polynomial-time verification process.

Findings

Extremality criteria are both necessary and sufficient.

Verification of extremality can be performed in $O(n^2)$ time.

Provides a practical method for analyzing solutions in max-plus algebra.

Abstract

We present necessary and sufficient criteria for a max-algebraic supereigenvector, i.e., a solution of the system $A\otimes\textbf{x}\geq\textbf{x}$ with $A\in\overline{\mathbb{R}}^{n\times n}$ in max-plus algebra, to be an extremal. We also show that the suggested extremality criteria can be verified in $O(n^2)$ time for any given solution $\textbf{x}$.