  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 $d>0$.
