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 $P e NP$, 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.
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