Section: New Results

Weak CSR Expansions and Transience Bounds in Max-Plus Algebra

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: $A^t=C{S}^{t}R\oplus B^t$. 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. As a result, we obtain a collection of bounds, in general incomparable to one another, but better than the bounds found in the literature.

This work was done by Thomas Nowak together with Glenn Merlet from Aix-Marseille Unversité and Sergeĭ Sergeev from the University of Birmingham. It appeared in the journal Linear Algebra and its Applications.