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.

中文翻译：

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NZ 矩阵原始集的 k 会合时间的线性边界

如果存在这些矩阵的乘积是入口正的，则一组非负矩阵称为原始矩阵。受最近有关同步自动机和原始集的结果的启发，我们研究了具有 k 个正条目的列或行的原始集的最短乘积的长度，称为其 k 交会时间 (k-RT)，在这种情况下没有零行和零列的矩阵集。我们证明 k-RT 至多与矩阵大小 n 相对于小 k 是线性的，而同步自动机的问题仍然存在。我们提供了 k-RT 的两个上限：第二个是第一个的改进，尽管后者可以写成封闭形式。然后，我们报告将 k-RT 的上限与启发式近似方法进行比较的数值结果。