  For general input automata, there exist regular constraint languages such
that asking if a given input automaton admits a synchronizing word in the
constraint language is PSPACE-complete or NP-complete. Here, we investigate
this problem for commutative automata over an arbitrary alphabet and automata
with simple idempotents over a binary alphabet as input automata. The latter
class contains, for example, the \v{C}ern\'y family of automata. We find that
for commutative input automata, the problem is always solvable in polynomial
time, for every constraint language. For input automata with simple idempotents
over a binary alphabet and with a constraint language given by a partial
automaton with up to three states, the constrained synchronization problem is
also solvable in polynomial time.

Formal Languages and Automata Theory;Computational Complexity