  In the Closest String problem one is given a family $\mathcal S$ of
equal-length strings over some fixed alphabet, and the task is to find a string
$y$ that minimizes the maximum Hamming distance between $y$ and a string from
$\mathcal S$. While polynomial-time approximation schemes (PTASes) for this
problem are known for a long time [Li et al., J. ACM'02], no efficient
polynomial-time approximation scheme (EPTAS) has been proposed so far. In this
paper, we prove that the existence of an EPTAS for Closest String is in fact
unlikely, as it would imply that $\mathrm{FPT}=\mathrm{W}[1]$, a highly
unexpected collapse in the hierarchy of parameterized complexity classes. Our
proof also shows that the existence of a PTAS for Closest String with running
time $f(\varepsilon)\cdot n^{o(1/\varepsilon)}$, for any computable function
$f$, would contradict the Exponential Time Hypothesis.
