Approximating the Maximum Number of Synchronizing States in Automata

TL;DR

This paper investigates the computational complexity and approximability of finding the largest synchronizing set in automata, establishing hardness results for various classes and polynomial solvability for unary automata.

Contribution

It proves the PSPACE-completeness of the Max Sync Set problem, derives inapproximability bounds for different automaton classes, and shows polynomial-time solvability for unary automata.

Findings

Max Sync Set decision problem is PSPACE-complete.

No polynomial-time approximation within certain factors unless P=NP.

Polynomial-time solution exists for unary 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 $\varepsilon > 0$, 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.