  We investigate the computational complexity of computing the Hausdorff
distance. Specifically, we show that the decision problem of whether the
Hausdorff distance of two semi-algebraic sets is bounded by a given threshold
is complete for the complexity class $\forall\exists_<\mathbb{R}$. This implies
that the problem is NP-, co-NP-, $\exists\mathbb{R}$- and
$\forall\mathbb{R}$-hard.
