The level set method for the two-sided max-plus eigenproblem

Authors:
Stéphane Gaubert;Sergeĭ Sergeev
Affiliations:
INRIA and Centre de Mathématiques Appliquées, École Polytechnique, Palaiseau Cédex, France 91128;School of Mathematics, University of Birmingham, Edgbaston, UK B15 2TT
Venue:
Discrete Event Dynamic Systems
Year:
2013

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Abstract

We consider the max-plus analogue of the eigenproblem for matrix pencils, A驴驴驴x驴=驴驴驴驴驴B驴驴驴x. We show that the spectrum of (A,B) (i.e., the set of possible values of 驴), which is a finite union of intervals, can be computed in pseudo-polynomial number of operations, by a (pseudo-polynomial) number of calls to an oracle that computes the value of a mean payoff game. The proof relies on the introduction of a spectral function, which we interpret in terms of the least Chebyshev distance between A驴驴驴x and 驴驴驴驴B驴驴驴x. The spectrum is obtained as the zero level set of this function.