The NP-completeness of the Road Coloring Problem

The Road Coloring Problem (RCP) originates in [1] and it was stated explicitly in the paper by Adler et al. [2]. It can be formulated as follows: let G be a strongly connected, constant out-degree finite digraph such that the greatest common divisor (gcd) of the lengths of all cycles in G equals 1. Is there an edge labeling, turning G into a deterministic finite synchronizing automaton? The problem is of great importance in automata theory, because the synchronizing property makes the automaton behavior resistant to errors that could occur in an input word: after the error is detected, the synchronizing word can reset the automaton to its initial state, as if there were no error. In this way we regain the control over automaton action. Trahtman [8] solved the RCP by showing that a synchronizing labeling exists for any strongly connected, constant out-degree finite digraph G if and only if the gcd of the lengths of all cycles in G equals 1. The RCP uses a notion of synchronization, which in fact was introduced few years before the work of Adler et al. The ‘classical’ version of the synchronizing problem (SP) is as follows: given a labeled graph G with constant out-degree, find the (shortest) synchronizing word for G . Natural questions about the complexity of