# A linear bound on the k-rendezvous time for primitive sets of NZ matrices

## TL;DR

This paper establishes a linear upper bound on the k-rendezvous time for primitive sets of nonnegative matrices with no zero rows or columns, advancing understanding of matrix product lengths related to automata synchronization.

## Contribution

It introduces the first linear bound for the k-rendezvous time in certain primitive matrix sets and provides improved upper bounds along with numerical comparisons.

## Key findings

- k-RT is at most linear in matrix size n for small k
- Two upper bounds on k-RT are proposed, with the second being an improvement
- Numerical results compare bounds with heuristic approximation methods

## 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, called its k-rendezvous time (k-RT}), in the case of sets of matrices having no zero rows and no zero columns. We prove that the k-RT is at most linear w.r.t. the matrix size n for small k, while the problem is still open for synchronizing automata. We provide two upper bounds on the k-RT: the second is an improvement of the first one, although the latter can be written in closed form. We then report numerical results comparing our upper bounds on the k-RT with heuristic approximation methods.