Strong Inapproximability of the Shortest Reset Word

The Černý conjecture states that every n-state synchronizing automaton has a reset word of length at most (n−1). We study the hardness of finding short reset words. It is known that the exact version of the problem, i.e., finding the shortest reset word, is NP-hard and coNP-hard, and complete for the DP class, and that approximating the length of the shortest reset word within a factor of O(log n) is NP-hard [Gerbush and Heeringa, CIAA’10], even for the binary alphabet [Berlinkov, DLT’13]. We significantly improve on these results by showing that, for every ε > 0, it is NP-hard to approximate the length of the shortest reset word within a factor of n1−ε. This is essentially tight since a simple O(n)-approximation algorithm exists.