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

Authors:
Vincent Blondel;Stéphane Gaubert;Natacha Portier
Affiliations:
Large Graphs and Networks, Département dingénierie mathématique, Université catholique de Louvain, 4 Avenue Georges Lemaítre, B-1348 Louvain-la-Neuve, Belgium;INRIA and CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France;LIP, UMR 5668, ENS de Lyon, CNRS, UCBL, INRIA, ícole Normale Supérieure de Lyon, Université de Lyon, 46, allée dItalie, 69364 Lyon cedex 07, France and Department of Computer S ...
Venue:
Journal of Computer and System Sciences
Year:
2011

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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 finite union of polyhedral sets.