# On the Tightness of Bounds for Transients and Weak CSR Expansions in Max-Plus Algebra

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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.
Document type :
Reports

https://hal.archives-ouvertes.fr/hal-01789012
Contributor : Thomas Nowak <>
Submitted on : Wednesday, May 9, 2018 - 4:13:34 PM
Last modification on : Thursday, January 23, 2020 - 6:22:13 PM

### Identifiers

• HAL Id : hal-01789012, version 1
• ARXIV : 1705.04104

### Citation

Glenn Merlet, Thomas Nowak, Sergei Sergeev. On the Tightness of Bounds for Transients and Weak CSR Expansions in Max-Plus Algebra. [Research Report] Arxiv. 2017. ⟨hal-01789012⟩

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