Strong, Strongly Universal and Weak Interval Eigenvectors in Max-Plus Algebra

Abstract

The optimization problems, such as scheduling or project management, in which the objective function depends on the operations maximum and plus, can be naturally formulated and solved in max-plus algebra. A system of discrete events, e.g., activations of processors in parallel computing, or activations of some other cooperating machines, is described by a systems of max-plus linear equations. In particular, if the system is in a steady state, such as a synchronized computer network in data processing, then the state vector is an eigenvector of the system. In reality, the entries of matrices and vectors are considered as intervals. The properties and recognition algorithms for several types of interval eigenvectors are studied in this paper. For a given interval matrix and interval vector, a set of generators is defined. Then, the strong and the strongly universal eigenvectors are studied and described as max-plus linear combinations of generators. Moreover, a polynomial recognition algorithm is suggested and its correctness is proved. Similar results are presented for the weak eigenvectors. The results are illustrated by numerical examples. The results have a general character and can be applied in every max-plus algebra and every instance of the interval eigenproblem.

Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

“…Similarly to [7,[9][10][11][12][13][14][15][16][17][20][21][22], by an interval of R(m, n), we mean a subset of R(m, n) of the form Y = (y ij ) for i ∈ M, j ∈ N, where each y ij is an arbitrary interval belonging to R. For each i, j we denote y ij := inf y ij and y ij = sup y ij . Then, we also have…”

“…Moreover, some of the equivalent conditions of strong versions of solvability can be checked in a polynomial number of arithmetic operations. Systems of two-sided (max, plus)-linear equations A ⊗ x = B ⊗ x have been studied by many authors [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The motivation for this research are papers [8,10].…”