[Paper Review] Computational Aspects of the Hausdorff Distance in Unbounded Dimension

Stefan König|arXiv (Cornell University)|Jan 7, 2014

Computational Geometry and Mesh Generation29 references5 citations

TL;DR

This paper investigates the computational complexity of the Hausdorff distance between polytopes in unbounded dimension, showing that it is polynomial-time solvable when both polytopes are in vertex presentation, but NP- and W[1]-hard when at least one is in halfspace presentation. It further presents a polynomial-time (3√d + 1)-approximation algorithm for the matching problem involving homothetic transformations, even under W[1]-hard evaluation conditions, and establishes a Helly-type theorem for convex bodies via an optimality criterion akin to John’s Theorem.

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.

Motivation & Objective

• To determine the computational complexity of computing the Hausdorff distance between two polytopes in arbitrary dimension.
• To analyze the hardness of the problem when polytopes are presented in halfspace or vertex form.
• To investigate the matching problem where one convex body is homothetically transformed to minimize Hausdorff distance to another.
• To derive polynomial-time approximation algorithms for the matching problem, especially when evaluation is hard.
• To establish a Helly-type theorem for convex bodies based on an optimality condition analogous to John’s Theorem.

Proposed method

• Classifies the complexity of Hausdorff distance computation based on polytope presentation: vertex (V) or halfspace (H).
• Uses second-order cone programming (SOCP) to model the matching problem and approximate the Hausdorff distance when both polytopes are in V-presentation.
• Applies the 'Reference Point Method' by replacing the diameter of H-polytopes with that of their axis-aligned bounding boxes to enable polynomial-time approximation.
• Derives performance guarantees using support functions and properties of the unit ball and polar bodies.
• Employs Carathéodory’s Theorem to reduce the number of extreme points needed in containment constraints.
• Establishes a Helly-type result by leveraging an optimality condition derived from John’s Theorem for convex bodies.

Experimental results

Research questions

• RQ1 Is the Hausdorff distance between two polytopes computable in polynomial time when both are in vertex presentation in unbounded dimension?
• RQ2 What is the computational complexity of computing the Hausdorff distance when at least one polytope is in halfspace presentation?
• RQ3 Can the matching problem—minimizing Hausdorff distance via homothetic transformation—be solved efficiently for polytopes in vertex presentation?
• RQ4 Is there a polynomial-time approximation algorithm for the matching problem when the evaluation problem is W[1]-hard due to halfspace presentation?
• RQ5 What is the structural relationship between optimal homothety and convex geometry, such as a Helly-type theorem?

Key findings

• When both polytopes are in vertex presentation, the Hausdorff distance can be computed in polynomial time for any norm.
• If at least one polytope is in halfspace presentation, computing the Hausdorff distance is NP-hard and W[1]-hard.
• For two vertex-presented polytopes, the matching problem under homothety is solvable in polynomial time in arbitrary dimension.
• Even when one polytope is in halfspace presentation, a (3√d + 1)-approximation of the optimal Hausdorff distance can be computed in polynomial time.
• A Helly-type theorem is established for convex bodies: if a set of convex bodies has a common homothetic transformation minimizing Hausdorff distance, then a small subset of size at most d(d + 2) suffices to determine the optimal transformation.
• The reference point method, using bounding box diameters instead of polytope diameters, yields a (3√d + 1)-approximation with polynomial-time computability.