Approximating the Maximum Number of Synchronizing States in Automata

We consider the problem 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 PSPACEcomplete. We prove that, assuming P 6= NP , for any ε > 0 the Max Sync Set problem cannot be approximated in polynomial time within a factor of O(k) for monotonic kstate automata with an alphabet of linear size, within a factor of O(k 1 2 −ε) for binary k-state automata, and within a factor of O(k 1 3 −ε) for binary monotonic k-state automata. Finally, we prove that for unary automata the problem becomes polynomially solvable.