Tropical patterns of matrices and the Gondran–Minoux rank function

Abstract

We introduce the notion of the tropical matrix pattern, which provides a powerful tool to investigate tropical matrices. The above approach is then illustrated by the application to the study of the properties of the Gondran–Minoux rank function. Our main result states that up to a multiplication of matrix rows by non-zero constants the Gondran–Minoux independence of the matrix rows and that of the rows of its tropical pattern are equivalent.We also present a number of applications of our main result. In particular, we show that the problem of checking whether the Gondran–Minoux rank of a matrix is less than a given positive integer can be solved in a polynomial time in the size of the matrix. Another consequence of our main result states that the tropical rank, trop(A), and the determinantal rank, d(A), of tropical matrices satisfy the following inequalities: trop(A)⩾GMr(A), d(A)⩾GMr(A), trop(A)⩾d(A)+23. As an important corollary of this result we obtain that if one of these functions is bounded then the other two are also bounded unlike the situation with the factor and Kapranov ranks.