Strong Inapproximability of the Shortest Reset Word

Abstract

The Cerný conjecture states that every n-state synchronizing automaton has a reset word of length at most \((n-1)^2\). 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 \(\mathrm {NP}\)-hard and \(\mathrm {coNP}\)-hard, and complete for the \(\mathrm {DP}\) class, and that approximating the length of the shortest reset word within a factor of \(O(\log n)\) is \(\mathrm {NP}\)-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 \(\varepsilon >0\), it is \(\mathrm {NP}\)-hard to approximate the length of the shortest reset word within a factor of \(n^{1-\varepsilon }\). This is essentially tight since a simple O(n)-approximation algorithm exists.