We are considering a two-stage optimal scheduling problem, which involves two similar projects with the same starting times for workers and the same deadlines for tasks. It is required that the starting times for workers and deadlines for tasks should be optimal for the first-stage project and, under this condition, also for the second-stage project. Optimality is measured with respect to the maximal lateness (or maximal delay) of tasks, which has to be minimized. We represent this problem as a...

A tropical version of Stickel's key exchange protocol was suggested by Grigoriev and Shpilrain (2014) and successfully attacked by Kotov and Ushakov (2018). We suggest some modifications of this scheme that use commuting matrices in tropical algebra and discuss some possibilities of attacks on these new modifications. We suggest some simple heuristic attacks on one of our new protocols, and then we generalize the Kotov and Ushakov attack on tropical Stickel's protocol and discuss the application...

The interval eigenproblem in max-min algebra is studied. A classification of interval eigenvectors is introduced and six types of interval eigenvectors are described. Characterization of all six types is given for the case of strictly increasing eigenvectors and Hasse diagram of relations between the types is presented.

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.

MM functions, formed by finite composition of the operators min, max and translation, represent discrete-event systems involving disjunction, conjunction and delay. The paper shows how they may be formulated as homogeneous rational algebraic functions of degree one, over (max, +) algebra, and reviews the properties of such homogeneous functions, illustrated by some orbit-stability problems.

This paper discusses the properties of reachability and observability for linear systems over the max-plus algebra. Working in the event-domain, the concept of asticity is used to develop conditions for weak reachability and weak observability. In the reachability problem, residuation is used to determine if a state is reachable and to generate the required control sequence to reach it. In the observability problem, residuation is used to estimate the state. Finally, as in the continuous-variable...

We propose an efficient method for finding a Chebyshev-best soluble approximation to an insoluble system of linear equations over max-plus algebra.

A finite iteration method for solving systems of (max, min)-linear equations is presented. The systems have variables on both sides of the equations. The algorithm has polynomial complexity and may be extended to wider classes of equations with a similar structure.

This paper demonstrates that the sensor cover energy problem in wireless communication can be transformed into a linear programming problem with max-plus linear inequality constraints. Consequently, by a well-developed preprocessing procedure, it can be further reformulated as a 0-1 integer linear programming problem and hence tackled by the routine techniques developed in linear and integer optimization. The performance of this two-stage solution approach is evaluated on a set of randomly generated...

In max-min algebra the standard pair of operations plus and times is replaced by the pair of operations maximum and minimum, respectively. A max-min matrix $A$ is called strongly robust if the orbit $x,A\otimes x,A^2\otimes x,\cdots$ reaches the greatest eigenvector with any starting vector. We study a special type of the strong robustness called the strong X-robustness, the case that a starting vector is limited by a lower bound vector and an upper bound vector. The equivalent condition for the strong X-robustness is introduced...

A matrix $A$ is said to have $𝐗$-simple image eigenspace if any eigenvector $x$ belonging to the interval $𝐗=\{x:\underline{x}\le x\le \overline{x}\}$ containing a constant vector is the unique solution of the system $A\otimes y=x$ in $𝐗$. The main result of this paper is an extension of $𝐗$-simplicity to interval max-min matrix $𝐀=\{A:\underline{A}\le A\le \overline{A}\}$ distinguishing two possibilities, that at least one matrix or all matrices from a given interval have $𝐗$-simple image eigenspace. $𝐗$-simplicity of interval matrices in max-min algebra are studied and equivalent conditions for interval...