Weak CSR expansions and transience bounds in max-plus algebra

Glenn Merlet, Thomas Nowak, Sergey Sergeev

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)
163 Downloads (Pure)


This paper aims to unify and extend existing techniques for deriving upper bounds on the transient of max-plus matrix powers. To this aim, we introduce the concept of weak CSR expansions: At=CStR⊕Bt. We observe that most of the known bounds (implicitly) take the maximum of (i) a bound for the weak CSR expansion to hold, which does not depend on the values of the entries of the matrix but only on its pattern, and (ii) a bound for the CStR term to dominate.

To improve and analyze (i), we consider various cycle replacement techniques and show that some of the known bounds for indices and exponents of digraphs apply here. We also show how to make use of various parameters of digraphs. To improve and analyze (ii), we introduce three different kinds of weak CSR expansions (named after Nachtigall, Hartman–Arguelles, and Cycle Threshold). As a result, we obtain a collection of bounds, in general incomparable to one another, but better than the bounds found in the literature.
Original languageEnglish
Pages (from-to)163-199
JournalLinear Algebra and its Applications
Early online date23 Aug 2014
Publication statusPublished - 15 Nov 2014


  • Max-plus
  • Matrix powers
  • Transient
  • Periodicity
  • Digraphs


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