  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.
