Eigenproblems in addition-min algebra
Meng Li, Xue-ping Wang
TL;DR
This work develops a comprehensive framework for eigenproblems in addition-min algebra motivated by P2P data transmission stability. It introduces a constructive representation and necessary/sufficient conditions for eigenvectors, and provides algorithms to compute all eigenvalues and eigenvectors, constrained eigenpairs, and supereigenpairs, with a unified combinatorial structure and a complexity bound of $O((n+1)^n)$. The constrained variant extends the theory to fuzzy relation inequalities with feasibility bounds, while the supereigenvariant analyzes maximum constrained supereigenvalues and the feasible region of corresponding vectors, offering practical mechanisms to study steady states and feasibility in fuzzy systems. Together, these contributions yield a principled, algorithmic toolkit for steady-state analysis in addition-min algebras with potential applications to network stability and quality-of-service problems in P2P settings.
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.