New transience bounds for max-plus linear systems

Abstract

We study the transients of linear max-plus dynamical systems. For that, we consider for each irreducible max-plus matrix A, the weighted graph G(A) such that A is the adjacency matrix of G(A). Based on a novel graph-theoretic counterpart to the number-theoretic Brauer's theorem, we propose two new methods for the construction of arbitrarily long paths in G(A) with maximal weight. That leads to two new upper bounds on the transient of a linear max-plus system which both improve on the bounds previously given by Even and Rajsbaum (STOC 1990, Theory of Computing Systems 1997, by Bouillard and Gaujal (Research Report 2000), and by Soto y Koelemeijer (PhD Thesis 2003), and are, in general, incomparable with Hartmann and Arguelles' bound (Mathematics of Operations Research 1999). With our approach, we also show how to improve the latter bound by a factor of two.A significant benefit of our bounds is that each of them turns out to be linear in the size of the system in various classes of linear max-plus system whereas the bounds previously given are all at least quadratic. Our second result concerns the relationship between matrix and system transients: We prove that the transient of an N × N matrix A is, up to some constant, equal to the transient of an A-linear system with an initial vector whose norm is quadratic in N . Finally, we study the applicability of our results to the well-known Full Reversal algorithm whose behavior can be described as a min-plus linear system.

Computation of the Transient in Max-Plus Linear Systems via SMT-Solving

“…If it is true that a j ∼ b then we get ( 6), otherwise we get (7). The proof for (8)-( 9) is similar to that of ( 6)- (7).…”

“…Even for a small n, the transient of A ∈ R n×n max can be large. Upper bounds of the transient have been discussed in [7,15,18,19].…”

“…Thus, it is possible for the resulting transient to be relatively large for a small-dimensional MPL system. There are several known upper bounds [7,15,18,19] for the transient, which are mostly computed via the corresponding precedence graph and are, in practice, much larger than the actual values.…”

Bounded Model Checking of Max-Plus Linear Systems via Predicate Abstractions

“…On the other hand, the transient tr(A) is harder to compute since it is unrelated to the dimension of A. The transient of A ∈ R n×n max can be really large even for a small n. Upper bounds for the transient have been discussed in [8,9,10,11].…”

“…Thus, it is possible for the resulting transient to be relatively large for a small-dimensional MPL system. There are several known upper bounds [8,9,10,11] for the transient, which are mostly computed via the corresponding precedence graph and are, in practice, much larger than the actual values. Hence, the ability to compute transients is important in practice.…”