The Complexity of the Hausdorff Distance

12/08/2021

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by   Paul Jungeblut, et al.

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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 ∀∃_<ℝ. This implies that the problem is NP-, co-NP-, ∃ℝ- and ∀ℝ-hard.