Complexity of Road Coloring with Prescribed Reset Words

By the Road Coloring Theorem (Trahtman, 2008), the edges of any aperiodic directed multigraph with a constant out-degree can be colored such that the resulting automaton admits a reset word. There may also be a need for a particular reset word to be admitted. For certain words it is NP-complete to decide whether there is a suitable coloring of a given multigraph. We present a classification of all words over the binary alphabet that separates such words from those that make the problem solvable in polynomial time. We show that the classification becomes different if we consider only strongly connected multigraphs. In this restricted setting the classification remains incomplete.Comment: To be presented at LATA 201

On Nonnegative Integer Matrices and Short Killing Words

Let $n$ be a natural number and $\mathcal{M}$ a set of $n \times n$-matrices over the nonnegative integers such that the joint spectral radius of $\mathcal{M}$ is at most one. We show that if the zero matrix $0$ is a product of matrices in $\mathcal{M}$, then there are $M_1, \ldots, M_{n^5} \in \mathcal{M}$ with $M_1 \cdots M_{n^5} = 0$. This result has applications in automata theory and the theory of codes. Specifically, if $X \subset \Sigma^*$ is a finite incomplete code, then there exists a word $w \in \Sigma^*$ of length polynomial in $\sum_{x \in X} |x|$ such that $w$ is not a factor of any word in $X^*$. This proves a weak version of Restivo's conjecture.Comment: This version is a journal submission based on a STACS'19 paper. It extends the conference version as follows. (1) The main result has been generalized to apply to monoids generated by finite sets whose joint spectral radius is at most 1. (2) The use of Carpi's theorem is avoided to make the paper more self-contained. (3) A more precise result is offered on Restivo's conjecture for finite code