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

Qianwei Zhuang

TL;DR

The problem addressed is exact convex-hull computation in high-dimensional spaces and identification of the extreme points $\mathcal{E}$. The authors propose an iterative approach that builds a compact reference set $\mathcal{R}_l$ per point and uses per-point quadratic-programming projections to drive $d(\boldsymbol{x}_l,\mathcal{R}_l)\to0$, yielding the final hull conv$(\mathcal{E})$. The main contributions are a polynomial-time algorithm with worst-case bound $O(n^{p+2}\log(1/\epsilon))$, formal convergence guarantees for all points, and dimension-agnostic scalability that overcomes the exponential-time limitations of classical methods. This approach enables exact hull recovery in high dimensions with potential practical benefits for geometric computation and related applications.

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 \( O(n^{p+2} \log(1/ε)) \), 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 \( ε\) 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.

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

TL;DR

The problem addressed is exact convex-hull computation in high-dimensional spaces and identification of the extreme points . The authors propose an iterative approach that builds a compact reference set per point and uses per-point quadratic-programming projections to drive , yielding the final hull conv. The main contributions are a polynomial-time algorithm with worst-case bound , formal convergence guarantees for all points, and dimension-agnostic scalability that overcomes the exponential-time limitations of classical methods. This approach enables exact hull recovery in high dimensions with potential practical benefits for geometric computation and related applications.

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 points in an -dimensional space, the algorithm achieves a dimension-independent worst-case time complexity of \( O(n^{p+2} \log(1/ε)) \), where depends on the choice of QP solver (e.g., corresponds to the worst-case bound when using an interior-point method), and 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.

Paper Structure

This paper contains 7 sections, 3 theorems, 54 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Convex Hull Problem
  3. Exact Convex Hull Construction
  4. Complexity Analysis
  5. Summary
  6. Proof of Lemma \ref{['lemma:extreme point']}
  7. Monotonic Distance Decrease

Key Result

Lemma 2.1

Assume $\boldsymbol{v} \in \mathbb{R}^m \setminus \{ \boldsymbol{0} \}$. For a point $\boldsymbol{x}_l \in \mathcal{A}$, if it uniquely satisfies then $\boldsymbol{x}_l$ is an extreme point of $\mathcal{A}$.

Figures (4)

  • Figure 1: The Exact Convex Hull
  • Figure 2: Partial Convex Hull Initialization
  • Figure 3: Implementation for Point $\mathsf{8}$
  • Figure 4: Monotonic Distance Decrease

Theorems & Definitions (4)

  • Definition 2.1
  • Lemma 2.1
  • Theorem 3.1
  • Corollary 3.1