Abstract
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.
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