Strong inapproximability of the shortest reset word

TL;DR

This paper proves that approximating the shortest reset word in automata within any factor better than roughly the automaton size to the power of one minus epsilon is NP-hard, significantly strengthening previous hardness results.

Contribution

It establishes a nearly tight NP-hardness of approximation for the shortest reset word problem, improving upon prior bounds and showing the problem's intractability for better approximations.

Findings

NP-hard to approximate within a factor of n^{1-ε} for any ε>0

Existing simple O(n)-approximation is essentially optimal

Strengthens previous hardness results for the problem

Abstract

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.