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

We consider the max-plus analogue of the eigenproblem for matrix pencils, A aSuaEuro parts per thousand x = lambda aSuaEuro parts per thousand B aSuaEuro parts per thousand x. We show that the spectrum of (A,B) (i.e., the set of possible values of lambda), 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 aSuaEuro parts per thousand x and lambda aSuaEuro parts per thousand B aSuaEuro parts per thousand x. The spectrum is obtained as the zero level set of this function.