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For general input automata, there exist regular constraint languages suchthat asking if a given input automaton admits a synchronizing word in theconstraint language is PSPACE-complete or NP-complete. Here, we investigatethis problem for commutative automata over an arbitrary alphabet and automatawith simple idempotents over a binary alphabet as input automata. The latterclass contains, for example, the \v{C}ern\'y family of automata. We find thatfor commutative input automata, the problem is always solvable in polynomialtime, for every constraint language. For input automata with simple idempotentsover a binary alphabet and with a constraint language given by a partialautomaton with up to three states, the constrained synchronization problem isalso solvable in polynomial time.

Constrained Synchronization for Commutative Automata and Automata with Simple Idempotents