On two Algorithmic Problems about Synchronizing Automata

We consider two basic problems arising in the theory of synchronizing automata: deciding, whether or not a given $n$-state automaton is synchronizing and the problem of approximating the reset threshold for a given synchronizing automaton. For the first problem of deciding whether or not a given $n$-state automaton is synchronizing we present an algorithm based on~\cite{RandSynch} with linear in $n$ expected time, while the best known algorithm is quadratic on each instance. For the second problem, we prove that unless \textsf{P} = \textsf{NP}, no polynomial time algorithm approximates the reset threshold for a given $n$-state $2$-letter automata within performance ratio less than $0.5 c \ln{(n)}$ where $c$ is a specific constant from~\cite{AMS6}. This improves the previous result of the author~\cite{MyTOCS2013} about non-approximability within any constant factor and also gives the positive answer to the corresponding conjecture from~\cite{Gerb1}.