Asymmetric Convex Intersection Testing

TL;DR

This paper introduces the asymmetric convex intersection testing problem, analyzing its complexity and proposing a simple, efficient deterministic algorithm with linear time dependence on input size, relevant for high-dimensional data analysis.

Contribution

It formulates the ACIT problem, connects it to existing LP-type problems, and presents a novel, simple deterministic algorithm with favorable runtime characteristics.

Findings

The problem of ACIT is identified as a natural, previously unexplored variant.

A straightforward deterministic algorithm for ACIT is developed.

The algorithm's runtime is linear in input size and depends reasonably on the dimension.

Abstract

We consider asymmetric convex intersection testing (ACIT). Let $P \subset \mathbb{R}^d$ be a set of $n$ points and $\mathcal{H}$ a set of $n$ halfspaces in $d$ dimensions. We denote by $\text{ch}(P)$ the polytope obtained by taking the convex hull of $P$, and by $\text{fh}(\mathcal{H})$ the polytope obtained by taking the intersection of the halfspaces in $\mathcal{H}$. Our goal is to decide whether the intersection of $\mathcal{H}$ and the convex hull of $P$ are disjoint. Even though ACIT is a natural variant of classic LP-type problems that have been studied at length in the literature, and despite its applications in the analysis of high-dimensional data sets, it appears that the problem has not been studied before. We discuss how known approaches can be used to attack the ACIT problem, and we provide a very simple strategy that leads to a deterministic algorithm, linear on $n$ and $m$, whose running time depends reasonably on the dimension $d$.