A polynomial algorithm for solving system of inequalities in max-plus algebra

This paper deals with the system of inequalities A⊗x⩾xA⊗x⩾x in which A   is a square matrix, xx is a column vector and ⊗⊗ is a max-plus composition. It provides two algorithms which can be used to find a finite proper supereigenvectors of A   corresponding to the eigenvalue λ=0λ=0 in a case of irreducible matrix A   with the maximum cycle mean λ(A)>0λ(A)>0 and a finite solution of the system in a case of reducible matrix A, respectively, in a polynomial number of steps.