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arXiv · 1705.04104

On the Tightness of Bounds for Transients of Weak CSR Expansions and Periodicity Transients of Critical Rows and Columns of Tropical Matrix Powers

Glenn Merlet·Thomas Nowak·Sergei Sergeev
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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 $\max\{T_1,T_2\}$, where $T_1$ is the weak CSR threshold and $T_2$ is the time after which the purely pseudoperiodic CSR terms start to dominate in the expansion. Various bounds have been derived for $T_1$ and $T_2$, 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 $T_1$, 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.

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Glenn Merlet, Thomas Nowak, Sergei Sergeev. 2017-05-11. On the Tightness of Bounds for Transients of Weak CSR Expansions and Periodicity Transients of Critical Rows and Columns of Tropical Matrix Powers. https://arxiv.org/abs/1705.04104

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