Weak CSR expansions and transience bounds in max-plus algebra

Abstract

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=CS^tR + 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 CS^tR 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.

32 pages

References in corpus (4)

• On visualisation scaling, subeigenvectors and Kleene stars in max algebra
• CSR expansions of matrix powers in max algebra
• New Transience Bounds for Long Walks
• On the Transience of Linear Max-Plus Dynamical Systems

Cited by in corpus (4)

• Semigroup identities of tropical matrices through matrix ranks
• On the tropical discrete logarithm problem and security of a protocol based on tropical semidirect product
• Algorithm for the CSR expansion of max-plus matrices using the characteristic polynomial
• New bounds on the periodicity transient of the powers of a tropical matrix: using cyclicity and factor rank