In this paper, we shall deal with the solvability of interval systems of linear equations in max-plus algebra. Max-plus algebra is an algebraic structure in which classical addition and multiplication are replaced by $\oplus$ and $\otimes$, where $a\oplus b=max\{a,b\}$, $a\otimes b=a+b$. The notation $𝔸\otimes x=𝕓$ represents an interval system of linear equations, where $𝔸=[\overline{b},\overline{A}]$ and $𝕓=[\underline{b},\overline{b}]$ are given interval matrix and interval vector, respectively. We can define several types of solvability of interval systems. In this paper, we define the T4 solvability and give an algorithm...

When a system of one-sided max-plus linear equations is inconsistent, the approximate solutions within an admissible error bound may be desired instead, particularly with some sparsity property. It is demonstrated in this paper that obtaining the sparsest approximate solution within a given $L_{\infty }$ error bound may be transformed in polynomial time into the set covering problem, which is known to be NP-hard. Besides, the problem of obtaining the sparsest approximate solution within a given $L_1$ error bound...

We consider the two-sided eigenproblem $A\otimes x=\lambda \otimes B\otimes x$ over max algebra. It is shown that any finite system of real intervals and points can be represented as spectrum of this eigenproblem.

We discuss the eigenvalue problem in the max-plus algebra. For a max-plus square matrix, the roots of its characteristic polynomial are not its eigenvalues. In this paper, we give the notion of algebraic eigenvectors associated with the roots of characteristic polynomials. Algebraic eigenvectors are the analogues of the usual eigenvectors in the following three senses: (1) An algebraic eigenvector satisfies an equation similar to the equation $A\otimes x=\lambda \otimes x$ for usual eigenvectors. Under a suitable assumption,...

A matrix $A$ in $(max,min)$-algebra (fuzzy matrix) is called weakly robust if $A^k\otimes x$ is an eigenvector of $A$ only if $x$ is an eigenvector of $A$. The weak robustness of fuzzy matrices are studied and its properties are proved. A characterization of the weak robustness of fuzzy matrices is presented and an $O\left(n^2\right)$ algorithm for checking the weak robustness is described.

In this paper we give a short, elementary proof of a known result in tropical mathematics, by which the convexity of the column span of a zero-diagonal real matrix $A$ is characterized by $A$ being a Kleene star. We give applications to alcoved polytopes, using normal idempotent matrices (which form a subclass of Kleene stars). For a normal matrix we define a norm and show that this is the radius of a hyperplane section of its tropical span.