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.

翻译：我们调查了计算Hausdorf距离的计算复杂性。 具体地说, 我们显示, 决定Hausdorf距离两组半高血压的距离是否受一个特定阈值约束的问题, 对于复杂等级$\ forall\ existences\\ mathbb{R} 来说是完全的。 这意味着问题在于 NP -, co- NP -, $\ divitys\ mathb{R} $- 和$\fall\ mathb{R}$- hard 。