A Linear Bound on the K-Rendezvous Time for Primitive Sets of NZ Matrices

Abstract

A set of nonnegative matrices is called primitive if there exists a product of these matrices that is entrywise positive. Motivated by recent results relating synchronizing automata and primitive sets, we study the length of the shortest product of a primitive set having a column or a row with k positive entries (the k-RT). We prove that this value is at most linear w.r.t. the matrix size n for small k, while the problem is still open for synchronizing automata. We then report numerical results comparing our upper bound on the k-RT with heuristic approximation methods.Keywords: Primitive set of matrices, synchronizing automaton, Černý conjecture.Deciding whether a set is primitive is an NP-hard problem if the set contains at least three matrices (while it is still of unknown complexity for sets of two matrices) [3], and so is computing its exponent. For sets of matrices having at least a positive entry in every row and every column (called NZ [13] or allowable matrices [16,18]), the primitivity problem becomes decidable in polynomial-time [28], although computing the exponent remains NP-hard [13]. Methods for approximating the exponent have been proposed [6] as well as upper bounds that depend just on the matrix size; in particular, if we denote with exp N Z (n) the maximal exponent among all the primitive sets of n × n NZ matrices, it is known that exp N Z (n) ≤ (15617n 3 + 7500n 2 + 56250n − 78125)/46875 [3,33]. Better upper bounds have been found for some classes of primitive sets [13,17]. The NZ condition is often met in applications and in particular in the connection with synchronizing automata.