Generalized Perron--Frobenius Theorem for Nonsquare Matrices

TL;DR

This paper extends the Perron--Frobenius theorem to nonsquare matrices, providing a theoretical foundation and polynomial-time algorithm for optimal solutions in systems like client-server models, with applications in wireless networks and economics.

Contribution

It introduces a generalized Perron--Frobenius theorem for nonsquare matrices and develops a polynomial-time algorithm to find optimal solutions in such settings.

Findings

Optimal solutions do not require cooperation among servers.

Choosing the best single server suffices for optimality.

The generalized theorem applies to power control and economic models.

Abstract

The celebrated Perron--Frobenius (PF) theorem is stated for irreducible nonnegative square matrices, and provides a simple characterization of their eigenvectors and eigenvalues. The importance of this theorem stems from the fact that eigenvalue problems on such matrices arise in many fields of science and engineering, including dynamical systems theory, economics, statistics and optimization. However, many real-life scenarios give rise to nonsquare matrices. A natural question is whether the PF Theorem (along with its applications) can be generalized to a nonsquare setting. Our paper provides a generalization of the PF Theorem to nonsquare matrices. The extension can be interpreted as representing client-server systems with additional degrees of freedom, where each client may choose between multiple servers that can cooperate in serving it (while potentially interfering with other clients). This formulation is motivated by applications to power control in wireless networks, economics and others, all of which extend known examples for the use of the original PF Theorem. We show that the option of cooperation between servers does not improve the situation, in the sense that in the optimal solution no cooperation is needed, and only one server needs to serve each client. Hence, the additional power of having several potential servers per client translates into \emph{choosing} the best single server and not into \emph{sharing} the load between the servers in some way, as one might have expected. The two main contributions of the paper are (i) a generalized PF Theorem that characterizes the optimal solution for a non-convex nonsquare problem, and (ii) an algorithm for finding the optimal solution in polynomial time.