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arXiv · 1903.10421

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

Costanza Catalano·Umer Azfar·Ludovic Charlier·Raphaël Jungers
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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.

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Costanza Catalano, Umer Azfar, Ludovic Charlier, Raphaël Jungers. 2019-03-25. A linear bound on the k-rendezvous time for primitive sets of NZ matrices. https://arxiv.org/abs/1903.10421

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