On the Coefficients of the Max-Algebraic Characteristic Polynomial and Equation

Abstract

No polynomial algorithms are known for finding the coefficients of the characteristic polynomial and characteristic equation of a matrix in max-algebra. The following are proved: (1) The task of finding the max-algebraic characteristic polynomial for permutation matrices is NP-complete. (2) The task of finding the lowest order finite term for a {0, -∞} matrix can be converted to the assignment problem. (3) The task of finding the max-algebraic characteristic equation of a {0, -∞} matrix can be converted to that of finding the conventional characteristic equation for a {0, 1} matrix and thus it is solvable in polynomial time.