On two Algorithmic Problems about Synchronizing Automata

Under the assumption \(\mathcal{P} \neq \mathcal{NP}\), we prove that two natural problems from the theory of synchronizing automata cannot be solved in polynomial time. The first problem is to decide whether a given reachable partial automaton is synchronizing. The second one is, given an \(n\)-state binary complete synchronizing automaton, to compute its reset threshold within performance ratio less than \(d \ln{(n)}\) for a specific constant \(d>0\).