TY - JOUR
T1 - Testing the irreducibility of nonsquare Perron-Frobenius systems
AU - Avin, C.
AU - Borokhovich, M.
AU - Haddad, Y.
AU - Kantor, E.
AU - Lotker, Z.
AU - Parter, M.
AU - Peleg, D.
N1 - Eshkol fellowship, Israel Ministry of Science and Technology; Israel Science Foundation [894/09]; Israel PBC [4/11]; Israel ISF [4/11]; United States-Israel Binational Science Foundation [2008348]; Israel Ministry of Science and Technology [6478]; Citigroup Foundation; Google Europe Fellowship; Google FellowshipSupported by Eshkol fellowship, the Israel Ministry of Science and Technology.Supported by a grant of the Israel Science Foundation.Supported in part by the Israel Science Foundation (grant 894/09), the I-CORE program of the Israel PBC and ISF (grant 4/11), the United States-Israel Binational Science Foundation (grant 2008348), the Israel Ministry of Science and Technology (infrastructures grant 6478), and the Citigroup Foundation.Recipient of the Google Europe Fellowship in distributed computing; research supported in part by this Google Fellowship.
PY - 2014/1/1
Y1 - 2014/1/1
N2 - The Perron-Frobenius (PF) theorem provides a simple characterization of the eigenvectors and eigenvalues of irreducible nonnegative square matrices. A generalization of the PF theorem to nonsquare matrices, which can be interpreted as representing systems with additional degrees of freedom, was recently presented in [1]. This generalized theorem requires a notion of irreducibility for nonsquare systems. A suitable definition, based on the property that every maximal square (legal) subsystem is irreducible, is provided in [1], and is shown to be necessary and sufficient for the generalized theorem to hold. This note shows that irreducibility of a nonsquare system can be tested in polynomial time. The analysis uses a graphic representation of the nonsquare system, termed the constraint graph, representing the flow of influence between the constraints of the system.
AB - The Perron-Frobenius (PF) theorem provides a simple characterization of the eigenvectors and eigenvalues of irreducible nonnegative square matrices. A generalization of the PF theorem to nonsquare matrices, which can be interpreted as representing systems with additional degrees of freedom, was recently presented in [1]. This generalized theorem requires a notion of irreducibility for nonsquare systems. A suitable definition, based on the property that every maximal square (legal) subsystem is irreducible, is provided in [1], and is shown to be necessary and sufficient for the generalized theorem to hold. This note shows that irreducibility of a nonsquare system can be tested in polynomial time. The analysis uses a graphic representation of the nonsquare system, termed the constraint graph, representing the flow of influence between the constraints of the system.
KW - Algorithms
KW - Irreducibility
KW - Perron-Frobenius theorem
UR - http://www.scopus.com/inward/record.url?scp=84904274056&partnerID=8YFLogxK
U2 - https://doi.org/10.1016/j.ipl.2014.05.004
DO - https://doi.org/10.1016/j.ipl.2014.05.004
M3 - Article
SN - 0020-0190
VL - 114
SP - 728
EP - 733
JO - Information Processing Letters
JF - Information Processing Letters
IS - 12
ER -