Peter Butkovič

Max-linear systems : theory and algorithms

Max-algebra: Two Special Features.- One-sided Max-linear Systems and Max-algebraic Subspaces.- Eigenvalues and Eigenvectors.- Maxpolynomials. The Characteristic Maxpolynomial.- Linear Independence and Rank. The Simple Image Set.- Two-sided Max-linear Systems.- Reachability of Eigenspaces.- Generalized Eigenproblem.- Max-linear Programs.- Conclusions and Open Problems.

Max-algebra: the linear algebra of combinatorics?

Abstract Let a ⊕ b =max( a , b ), a ⊗ b = a + b for a,b∈ R := R ∪{−∞} . By max-algebra we understand the analogue of linear algebra developed for the pair of operations (⊕,⊗) extended to matrices and vectors. Max-algebra, which has been studied for more than 40 years, is an attractive way of describing a class of nonlinear problems appearing for instance in machine-scheduling, information technology and discrete-event dynamic systems. This paper focuses on presenting a number of links between basic max-algebraic problems like systems of linear equations, eigenvalue–eigenvector problem, linear independence, regularity and characteristic polynomial on one hand and combinatorial or combinatorial optimisation problems on the other hand. This indicates that max-algebra may be regarded as a linear-algebraic encoding of a class of combinatorial problems. The paper is intended for wider readership including researchers not familiar with max-algebra.

Generators, extremals and bases of max cones

Abstract Max cones are max-algebraic analogs of convex cones. In the present paper we develop a theory of generating sets and extremals of max cones in R + n . This theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors. We give new proofs of existing results suitably generalizing, restating and refining them. Of these, it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements. We include results on properties of open and closed cones, on properties of totally dependent sets and on computational bounds for the problem of finding the (essentially unique) basis of a finitely generated cone.

Strong regularity of matrices: a survey of results

Abstract Let G = (G, ⊗, ≤) be a linearly ordered, commutative group and u⊕v = max(u, v) for all u, v ϵ G. Extend ⊕, ⊗ in the usual way on matrices over G. An m × n matrix A is said to have strongly linearly independent (SLI) columns, if for some b the system of equations A⊗x = b has a unique solution. If, moreover, m = n then A is said to be strongly regular (SR). This paper is a survey of results concerning SLI and SR with emphasis on computational complexity. We present also a similar theory developed for a structure based on a linearly ordered set where ⊕ is maximum and ⊗ is minimum.

The equation A ⊗ x = B ⊗ y over (max, +)

For the two-sided homogeneous linear equation system A ⊗ x = B ⊗ y over (max, +), with no infinite rows or columns in A or B, an algorithm is presented which converges to a finite solution from any finite starting point whenever a finite solution exists. If the finite elements of A, B are all integers, convergence is in a finite number of steps, for which a precise bound can be calculated if moreover one of A, B has only finite elements. The algorithm is thus pseudopolynomial in complexity.

On visualization scaling, subeigenvectors and Kleene stars in max algebra

Abstract The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is that of strict visualization scaling, defined as, for a given nonnegative matrix A , a diagonal matrix X such that all elements of X - 1 AX are less than or equal to the maximum cycle geometric mean of A , with strict inequality for the entries which do not lie on critical cycles. In this paper such scalings are described by means of the max algebraic subeigenvectors and Kleene stars of nonnegative matrices as well as by some concepts of convex geometry.

Minimal (max,+) Realization of Convex Sequences

We show that the minimal dimension of a linear realization over the (max,+) semiring of a convex sequence is equal to the minimal size of a decomposition of the sequence as a supremum of discrete affine maps. The minimal-dimensional realization of any convex realizable sequence can thus be found in linear time. The result is based on a bound in terms of minors of the Hankel matrix.

Reducible Spectral Theory with Applications to the Robustness of Matrices in Max-Algebra

Let

A condition for the strong regularity of matrices in the minimax algebra

a\oplus b=\max(a,b)

Simple image set of (max, +) linear mappings

and