  The \v{C}ern\'y 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 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 $\epsilon>0$, it is NP-hard
to approximate the length of the shortest reset word within a factor of
$n^{1-\epsilon}$. This is essentially tight since a simple $O(n)$-approximation
algorithm exists.
