Eigenproblems in addition-min algebra

TL;DR

This paper explores eigenproblems in addition-min algebra, providing conditions and algorithms for eigenvalues, eigenvectors, constrained eigenvectors, and supereigenvalues, with applications to data transmission stability.

Contribution

It introduces new necessary and sufficient conditions and algorithms for eigenproblems in addition-min algebra, including constrained and supereigenvalue problems.

Findings

Algorithms for eigenvalues and eigenvectors are developed.

Conditions for constrained eigenvectors are established.

Maximum constrained supereigenvalue is computed and illustrated.

Abstract

In order to guarantee the downloading quality requirements of users and improve the stability of data transmission in a BitTorrent-like peer-to-peer file sharing system, this article deals with eigenproblems of addition-min algebras. First, it provides a sufficient and necessary condition for a vector being an eigenvector of a given matrix, and then presents an algorithm for finding all eigenvalues and eigenvectors of a given matrix. It further proposes a sufficient and necessary condition for a vector being a constrained eigenvector of a given matrix and supplies an algorithm for computing all the constrained eigenvectors and eigenvalues of a given matrix. This article finally discusses the supereigenproblem of a given matrix and presents an algorithm for obtaining the maximum constrained supereigenvalue and depicting the feasible region of all the constrained supereigenvectors for a given matrix. It also gives some examples for illustrating the algorithms, respectively.