The Complexity of the Hausdorff Distance

TL;DR

This paper analyzes the computational complexity of calculating the Hausdorff distance between semi-algebraic sets, revealing it is complete for a complex class involving quantifiers over real numbers, indicating high computational difficulty.

Contribution

It establishes the decision problem for the Hausdorff distance as complete for the class _<, showing its computational hardness.

Findings

The problem is _<-hard.

It is NP-, co-NP-, - and -hard.

The problem's complexity class indicates high computational difficulty.

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.