Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums

TL;DR

This paper introduces efficient approximation algorithms for detecting intersections and computing Minkowski sums of multiple convex polytopes in high dimensions, with applications to width approximation of point sets.

Contribution

It presents novel data structures and algorithms for approximate intersection detection and Minkowski sum computation for multiple convex bodies in high dimensions.

Findings

Polytope intersection can be approximated in polylogarithmic query time.

Minkowski sum approximation is achieved in near-linear time with respect to input size and approximation parameter.

Width approximation of point sets is significantly improved using these new methods.

Abstract

Approximation problems involving a single convex body in $d$-dimensional space have received a great deal of attention in the computational geometry community. In contrast, works involving multiple convex bodies are generally limited to dimensions $d \leq 3$ and/or do not consider approximation. In this paper, we consider approximations to two natural problems involving multiple convex bodies: detecting whether two polytopes intersect and computing their Minkowski sum. Given an approximation parameter $\varepsilon > 0$, we show how to independently preprocess two polytopes $A,B$ into data structures of size $O(1/\varepsilon^{(d-1)/2})$ such that we can answer in polylogarithmic time whether $A$ and $B$ intersect approximately. More generally, we can answer this for the images of $A$ and $B$ under affine transformations. Next, we show how to $\varepsilon$-approximate the Minkowski sum of two given polytopes defined as the intersection of $n$ halfspaces in $O(n \log(1/\varepsilon) + 1/\varepsilon^{(d-1)/2 + \alpha})$ time, for any constant $\alpha > 0$. Finally, we present a surprising impact of these results to a well studied problem that considers a single convex body. We show how to $\varepsilon$-approximate the width of a set of $n$ points in $O(n \log(1/\varepsilon) + 1/\varepsilon^{(d-1)/2 + \alpha})$ time, for any constant $\alpha > 0$, a major improvement over the previous bound of roughly $O(n + 1/\varepsilon^{d-1})$ time.