Computational Aspects of the Hausdorff Distance in Unbounded Dimension

TL;DR

This paper explores the computational complexity of calculating the Hausdorff distance between polytopes in high dimensions and presents algorithms for approximating solutions, including transformations to minimize this distance.

Contribution

It provides a complexity analysis and polynomial-time approximation algorithms for Hausdorff distance problems involving polytopes in arbitrary dimensions.

Findings

Polynomial-time algorithms for approximating Hausdorff distance between polytopes.

Characterization of optimal solutions for homothetic transformations.

A Helly-type theorem related to the problem.

Abstract

We study the computational complexity of determining the Hausdorff distance of two polytopes given in halfspace- or vertex-presentation in arbitrary dimension. Subsequently, a matching problem is investigated where a convex body is allowed to be homothetically transformed in order to minimize its Hausdorff distance to another one. For this problem, we characterize optimal solutions, deduce a Helly-type theorem and give polynomial time (approximation) algorithms for polytopes.