5,927 research outputs found
The level set method for the two-sided eigenproblem
We consider the max-plus analogue of the eigenproblem for matrix pencils
Ax=lambda Bx. 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 Ax and lambda Bx. The spectrum is
obtained as the zero level set of this function.Comment: 34 pages, 4 figures. Changes with respect to the previous version: we
explain relation to mean-payoff games and discrete event systems, and show
that the reconstruction of spectrum is pseudopolynomia
On Integer Images of Max-plus Linear Mappings
Let us extend the pair of operations (max,+) over real numbers to matrices in
the same way as in conventional linear algebra. We study integer images of
max-plus linear mappings. The question whether Ax (in the max-plus algebra) is
an integer vector for at least one x has been studied for some time but
polynomial solution methods seem to exist only in special cases. In the
terminology of combinatorial matrix theory this question reads: is it possible
to add constants to the columns of a given matrix so that all row maxima are
integer? This problem has been motivated by attempts to solve a class of
job-scheduling problems. We present two polynomially solvable special cases
aiming to move closer to a polynomial solution method in the general case
Tropical linear algebra with the Lukasiewicz T-norm
The max-Lukasiewicz semiring is defined as the unit interval [0,1] equipped
with the arithmetics "a+b"=max(a,b) and "ab"=max(0,a+b-1). Linear algebra over
this semiring can be developed in the usual way. We observe that any problem of
the max-Lukasiewicz linear algebra can be equivalently formulated as a problem
of the tropical (max-plus) linear algebra. Based on this equivalence, we
develop a theory of the matrix powers and the eigenproblem over the
max-Lukasiewicz semiring.Comment: 27 page
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