The purpose of this paper is to study the tropical algebra – the algebra over the tropical semi-ring. We start by introducing a third approach to arithmetic over the max-plus semi-ring which generalizes the two other concepts in use. Regarding this new arithmetic, matters of tropical matrices are discussed and the properties of these matrices are studied. These are the preceding phases toward t...

The minimal realization problem in the max-plus algebra, " Chapter 32 in Open Problems in Mathematical Systems and Control Theory

A large class of dynamic systems-such as the material flow in production or assembly lines, the message flow in communication networks, and jobs in computer systems-can be modeled by Discrete Event Dynamic Systems (DEDS). These are systems where the occurrence of events is determined by the system itself and not described by time. Examples of such events are the beginning or completion of a tas...

For a rank-1 matrix A = a ⊗ b over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T (A) = U ⊗ A ⊗ V , or T (A) = U ⊗ A ⊗ V with ...

In this paper we give a survey of our research on the Extended Linear Complemen-tarity Problem (ELCP). First we discuss the link between the ELCP and other generalizations of the Linear Complementarity Problem, and we present an algorithm to nd all the solutions of an ELCP. Next we introduce the max-plus algebra and show how it can be used to model a certain class of discrete event systems. Fin...

This paper surveys complexity and approximability results for the Maximum Solution (Max Sol) problem. Max Sol is an optimisation variant of the constraint satisfaction problem. Many important and well-known combinatorial optimisation problems are instances of Max Sol: for example, Max Sol restricted to the domain {0, 1} is exactly the Max Ones problem (which, in turn, captures problems such as ...

We describe the specialization to max-plus algebra of Howard’s policy improvement scheme, which yields an algorithm to compute the solutions of spectral problems in the max-plus semiring. Experimentally, the algorithm shows a remarkable (almost linear) average execution time. Résumé: Nous spécialisons à l’algèbre max-plus l’itération sur les politiques de Howard, qui fournit un algorithme pour ...

We introduce a notion of dimension of max-min convex sets, following the approach of tropical convexity. We introduce a max-min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope. We describe the relation between this rank and the notion of strong regularity in max-min algebra, which is traditionally defined in terms of unique solvabi...

Automata with weights (multiplicities) in (max,+) algebra form a class of timed automata. Determinism is a crucial property for numerous results on (max,+) automata and, in particular, for applications to performance evaluation and control of a large class of timed discrete event systems. In this paper, we show how to build a deterministic (max,+) automaton equivalent to a live and safe timed P...

Linear systems over the max-plus algebra provide a suitable formalism to model discrete event where synchronization, without competition, is involved. In this paper, we consider formulation of matching problem for such class, in which output given system, called plant, forced, by input, track exactly that model. A necessary and sufficient condition its solvability obtained making use geometric ...