The set of realizations of a max-plus linear sequence is semi-polyhedral

TL;DR

This paper demonstrates that the set of all realizations of a max-plus linear sequence can be represented as a finite union of polyhedral sets, providing a computational approach to the realization problem.

Contribution

It introduces a method to characterize the realization set of max-plus linear sequences as semi-polyhedral, extending to rational expressions over semirings.

Findings

The realization set is a finite union of polyhedral sets.

An algorithm is provided to compute realizations from any given realization.

The results extend to rational expressions over idempotent semirings.

Abstract

We show that the set of realizations of a given dimension of a max-plus linear sequence is a finite union of polyhedral sets, which can be computed from any realization of the sequence. This yields an (expensive) algorithm to solve the max-plus minimal realization problem. These results are derived from general facts on rational expressions over idempotent commutative semirings: we show more generally that the set of values of the coefficients of a commutative rational expression in one letter that yield a given max-plus linear sequence is a semi-algebraic set in the max-plus sense. In particular, it is a finite union of polyhedral sets.