Abstract
We study the transients of matrices in max-plus algebra. Our approach is based on the weak CSR expansion. Using this expansion, the transient can be expressed by max{T1, T2}, where T1 is the weak CSR threshold and T2 is the time after which the purely pseudoperiodic CSR terms start to dominate in the expansion. Various bounds have been derived for T1 and T2, naturally leading to the question which matrices, if any, attain these bounds. In the present paper, we characterize the matrices attaining two particular bounds on T1, which are generalizations of the bounds of Wielandt and Dulmage–Mendelsohn on the indices of non-weighted digraphs. This also leads to a characterization of tightness for the same bounds on the transients of critical rows and columns. The characterizations themselves are generalizations of those for the non-weighted case.
Acknowledgments
We are grateful to Stéphane Gaubert and anonymous referees of the paper for many constructive and helpful suggestions and comments.
Disclosure statement
No potential conflict of interest was reported by the author(s).