An Efficient Algorithm for Vertex Enumeration of Arrangement

TL;DR

This work introduces the Zero rule, a novel pivot rule for vertex enumeration in hyperplane arrangements that eliminates the objective function, guarantees a unique terminal dictionary, and ensures termination within at most $d$ pivot steps. By integrating the Zero rule into Avis–Fukuda’s framework, the authors obtain a VE algorithm with complexity $O(n^2 d^2 (v-v_d) + nd v_d)$, where $v$ is the number of vertices and $v_d$ counts dictionaries requiring exactly $d$ steps; this yields $O(nd^4 v)$ in simple arrangements, a substantial improvement over existing methods. The approach is backed by a thorough complexity analysis and extensive experiments showing robust performance across diverse arrangements, outperforming Moss and the original/Enhanced AF in many regimes. The work contributes a principled, pivot-rule-driven VE method that leverages dictionary structure to achieve near-optimal pivot depth and reduced storage, with practical impact for computational geometry and related applications.

Abstract

This paper presents a state-of-the-art algorithm for the vertex enumeration problem of arrangements, which is based on the proposed new pivot rule, called the Zero rule. The Zero rule possesses several desirable properties: i) It gets rid of the objective function; ii) Its terminal satisfies uniqueness; iii) We establish the if-and-only if condition between the Zero rule and its valid reverse, which is not enjoyed by earlier rules; iv) Applying the Zero rule recursively definitely terminates in $d$ steps, where $d$ is the dimension of input variables. Because of so, given an arbitrary arrangement with $v$ vertices of $n$ hyperplanes in $\mathbb{R}^d$, the algorithm's complexity is at most $\mathcal{O}(n^2d^2v)$ and can be as low as $\mathcal{O}(nd^4v)$ if it is a simple arrangement, while Moss' algorithm takes $\mathcal{O}(nd^2v^2)$, and Avis and Fukuda's algorithm goes into a loop or skips vertices because the if-and-only-if condition between the rule they chose and its valid reverse is not fulfilled. Systematic and comprehensive experiments confirm that the Zero rule not only does not fail but also is the most efficient.

Figures (13)

• Figure 1: A dictionary with and without the objective function.
• Figure 2: An exemplary illustration of the Zero rule and its valid reverse. Blocks of different colors represent where different sub-conditions are enforced.
• Figure 3: Zeros in the row of $x_{n_j}$ on $\mathbf{x}_{\tilde{B}}=\tilde{A}\mathbf{x}_{\tilde{N}}$.
• Figure 4: An example of the lex-min test. LHS: fails the test. RHS: passes the test.
• Figure 5: The process of checking and performing valid reverse pivots. LHS: Avis and Fukuda avis1991pivoting; RHS: Our proposed algorithm.

Theorems & Definitions (24)

• Definition 1: Hyperplane and arrangement
• Definition 2: Vertex of an arrangement
• Definition 3: Slack variable
• Definition 4: Dictionary with an objective function
• Definition 5: Dictionary and Basic solution
• Definition 6: Basis and co-basis
• Definition 7: Dictionary without the objective function
• Definition 8: Primal Feasible
• Definition 9: Dual Feasible and Optimal
• Definition 10: Pivoting rule