Approximating the Maximum Number of Synchronizing States in Automata

Abstract

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 $\epsilon >0$, the {\sc Max Sync Set} problem cannot be approximated in polynomial time within a factor of $O(n^{1-\epsilon })$ for weakly acyclic $n$-state automata with alphabet of linear size, within a factor of $O(n^{\frac{1}{2}-\epsilon })$ for binary $n$-state automata, and within a factor of $O(n^{\frac{1}{3}-\epsilon })$ for binary weakly acyclic $n$-state automata. Finally, we prove that for unary automata the problem becomes solvable in polynomial time.

8 pages, 1 figure; a mistake in the class of automata in the main result is fixed