Generalized Perron-Frobenius Theorem for Nonsquare Matrices

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. Despite the extensive development of spectral theories for nonnegative matrices, the applicability of such theories to non-convex optimization problems is not clear. In particular, 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 ∗Ben Gurion University, Beer-Sheva, Israel. Email: {avin, borokhom, zvilo}@cse.bgu.ac.il, [email protected]. †Jerusalem College of Technology, Jerusalem, Israel. ‡The Technion, Haifa, Israel. Email: [email protected]. Supported by Eshkol fellowship, the Israel Ministry of Science and Technology. §Supported by a grant of the Israel Science Foundation. ¶The Weizmann Institute of Science, Rehovot, Israel. Email: {merav.parter,david.peleg}@ weizmann.ac.il. ‖Recipient of the Google Europe Fellowship in distributed computing; research supported in part by this Google Fellowship. ∗∗Supported in part by the Israel Science Foundation (grant 894/09), the United States-Israel Binational Science Foundation (grant 2008348), the I-CORE program of the Israel PBC and ISF (grant 4/11), the Israel Ministry of Science and Technology (infrastructures grant), and the Citi Foundation. 1 ar X iv :1 30 8. 59 15 v1 [ cs .N A ] 2 7 A ug 2 01 3 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 choosing the best single server and not into 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. Towards achieving those goals, we extend the definitions of irreducibility and largest eigenvalue of square matrices to nonsquare ones in a novel and non-trivial way, which turns out to be necessary and sufficient for our generalized theorem to hold. The analysis performed to characterize the optimal solution uses techniques from a wide range of areas and exploits combinatorial properties of polytopes, graph-theoretic techniques and analytic tools such as spectral properties of nonnegative matrices and root characterization of integer polynomials.