A Polynomial-Time Algorithm for Computing the Exact Convex Hull in High-Dimensional Spaces

TL;DR

This paper introduces a polynomial-time algorithm for exactly computing the convex hull in high-dimensional spaces, overcoming the limitations of existing exponential-time methods and providing theoretical guarantees.

Contribution

The paper presents a novel iterative quadratic programming-based algorithm with dimension-independent polynomial complexity for exact convex hull computation in high dimensions.

Findings

Achieves polynomial-time complexity for high-dimensional convex hulls.

Provides theoretical guarantees for exact hull identification.

Demonstrates efficiency in high-dimensional datasets.

Abstract

This study presents a novel algorithm for identifying the set of extreme points that constitute the exact convex hull of a point set in high-dimensional Euclidean space. The proposed method iteratively solves a sequence of dynamically updated quadratic programming (QP) problems for each point and exploits their solutions to provide theoretical guarantees for exact convex hull identification. For a dataset of \( n \) points in an \( m \)-dimensional space, the algorithm achieves a dimension-independent worst-case time complexity of \( O(n^{p+2} \log(1/\epsilon)) \), where \( p \) depends on the choice of QP solver (e.g., \( p = 4 \) corresponds to the worst-case bound when using an interior-point method), and \( \epsilon \) denotes the target numerical precision (i.e., the optimality tolerance of the QP solver). The proposed method is applicable to spaces of arbitrary dimensionality and exhibits particular efficiency in high-dimensional settings, owing to its polynomial-time complexity, whereas existing exponential-time algorithms become computationally impractical.