Algorithms for Deciding Membership in Polytopes of General Dimension

TL;DR

This paper introduces an efficient, scalable algorithm for polytope membership testing in high dimensions by reducing the problem to approximate nearest neighbor search, enabling practical handling of large convex polytopes.

Contribution

The paper presents a novel reduction of polytope membership to ANN, along with a boundary data structure and implementation that outperform brute-force methods in high-dimensional settings.

Findings

Algorithm scales well with dimension and number of facets

Reduction to ANN enables polynomial complexity bounds

Implementation outperforms brute-force approaches

Abstract

We study the fundamental problem of polytope membership aiming at large convex polytopes, i.e. in high dimension and with many facets, given as an intersection of halfspaces. Standard data-structures as well as brute force methods cannot scale, due to the curse of dimen- sionality. We design an efficient algorithm, by reduction to the approx- imate Nearest Neighbor (ANN) problem based on the construction of a Voronoi diagram with the polytope being one bounded cell. We thus trade exactness for efficiency so as to obtain complexity bounds polyno- mial in the dimension, by exploiting recent progress in the complexity of ANN search. We employ this algorithm to present a novel boundary data structure based on a Newton-like iterative intersection procedure. We implement our algorithms and compare with brute-force approaches to show that they scale very well as the dimension and number of facets grow larger.