Approximating the Maximum Number of Synchronizing States in Automata

We consider the problem {\sc Max Sync Set} of finding a maximum synchronizing set of states in a given automaton. We show that the decision version of this problem is PSPACE-complete and investigate the approximability of {\sc Max Sync Set} for binary and weakly acyclic automata (an automaton is called weakly acyclic if it contains no cycles other than self-loops). We prove that, assuming $P \ne NP$, for any $\varepsilon > 0$, the {\sc Max Sync Set} problem cannot be approximated in polynomial time within a factor of $O(n^{1 - \varepsilon})$ for weakly acyclic $n$-state automata with alphabet of linear size, within a factor of $O(n^{\frac{1}{2} - \varepsilon})$ for binary $n$-state automata, and within a factor of $O(n^{\frac{1}{3} - \varepsilon})$ for binary weakly acyclic $n$-state automata. Finally, we prove that for unary automata the problem becomes solvable in polynomial time.