Efficient algorithms for deciding the type of growth of products of integer matrices

For a given finite set $\Sigma$ of matrices with nonnegative integer entries we study the growth of $$ \max_t(\Sigma) = \max\{\|A_{1}... A_{t}\|: A_i \in \Sigma\}.$$ We show how to determine in polynomial time whether the growth with $t$ is bounded, polynomial, or exponential, and we characterize precisely all possible behaviors.