  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.
