Approximating the minimum length of synchronizing words is hard

TL;DR

This paper proves that, assuming P≠NP, there is no polynomial-time algorithm capable of approximating the shortest synchronizing word length within any constant factor, highlighting the problem's computational hardness.

Contribution

It establishes the hardness of approximating the minimum length of synchronizing words within a constant factor under standard complexity assumptions.

Findings

No polynomial algorithm can approximate within a constant factor unless P=NP.

The problem is computationally hard to approximate.

Provides theoretical bounds on the difficulty of synchronization problems.

Abstract

We prove that, unless $\mathrm{P}=\mathrm{NP}$, no polynomial algorithm can approximate the minimum length of \sws for a given \san within a constant factor.