Approximate Vertex Enumeration

TL;DR

This work introduces the problem of approximate vertex enumeration, where one seeks an $\varepsilon$-approximate V-representation of a polytope $P$ given by $Ax\le \mathbbm{1}$, satisfying $P \subseteq Q \subseteq (1+\varepsilon)P$ for $Q=\mathrm{conv}(\mathcal{V})$. It presents two algorithms—the approximate double description method (ADDM) and the shortcut algorithm (SCA)—and proves correctness for dimensions $d\in\{2,3\}$, including stability under imprecise arithmetic when errors are bounded. The methods differ in approach: ADDM uses a graph-based refinement of the vertex set, while SCA constructs a planar graph and uses bounding walks to generate a valid subset of vertices; crucially, SCA’s planarity enables a direct correctness proof in 2D/3D and implies ADDM’s correctness in these dimensions. Numerical results indicate that the approximate problem can be easier to solve than the exact one, with larger tolerances reducing the number of vertices, though higher dimensions ($d\ge 4$) remain challenging with open questions about algorithmic reliability and necessary assumptions. Overall, the work lays a foundation for stable, approximate convex-geometry computations and highlights open problems in extending guarantees beyond dimension 3.

Abstract

The problem to compute the vertices of a polytope given by affine inequalities is called vertex enumeration. The inverse problem, which is equivalent by polarity, is called the convex hull problem. We introduce `approximate vertex enumeration' as the problem to compute the vertices of a polytope which is close to the original polytope given by affine inequalities. In contrast to exact vertex enumerations, both polytopes are not required to be combinatorially equivalent. Two algorithms for this problem are introduced. The first one is an approximate variant of Motzkin's double description method. Only under certain strong conditions, which are not acceptable for practical reasons, we were able to prove correctness of this method for polytopes of arbitrary dimension. The second method, called shortcut algorithm, is based on constructing a plane graph and is restricted to polytopes of dimension 2 and 3. We prove correctness of the shortcut algorithm. As a consequence, we also obtain correctness of the approximate double description method, only for dimension 2 and 3 but without any restricting conditions as still required for higher dimensions. We show that for dimension 2 and 3 both algorithm remain correct if imprecise arithmetic is used and the computational error caused by imprecision is not too high. Both algorithms were implemented. The numerical examples motivate the approximate vertex enumeration problem by showing that the approximate problem is often easier to solve than the exact vertex enumeration problem. It remains open whether or not the approximate double description method (without any restricting condition) is correct for polytopes of dimension 4 and higher.

Figures (15)

• Figure 1: Algorithm \ref{['alg_basic_cut']} can fail, even in the plane. Left: $P= \mathop{\mathrm{conv}}\nolimits\{u_3,u_5,u_6\}$, $(1+\varepsilon)P= \mathop{\mathrm{conv}}\nolimits\{\bar{u}_3,\bar{u}_5,\bar{u}_6\}$ ($\varepsilon = 1$) and a new inequality, illustrated by $H_0$. The set $\mathcal{V}=\{v_1,v_3,v_5,v_2\}$ provides an $\varepsilon$-approximate V-representation of $P$. Right: Result of Algorithm \ref{['alg_basic_cut']}. It computes $\mathcal{V}'=\{v_1,v_3,v_7\}$, which is not an $\varepsilon$-approximate V-representation of $P'$ since $P' = \mathop{\mathrm{conv}}\nolimits\{u_3,u_7,u_8,u_6\} \not\subseteq \mathop{\mathrm{conv}}\nolimits \mathcal{V}'$.
• Figure 2: $P= \mathop{\mathrm{conv}}\nolimits\{u_1,u_3,u_4,u_2\}$, $(1+\varepsilon)P= \mathop{\mathrm{conv}}\nolimits\{\bar{u}_1,\bar{u}_3,\bar{u}_4,\bar{u}_2\}$ and a "cut" $H_0$ are shown on the left. $\mathcal{V}=\{v_1,v_3,v_4,v_2\}$ is a strong $\varepsilon$-approximate V-representation of $P$. Algorithm \ref{['alg_basic_cut']} computes $\mathcal{V}'=\{v_1,v_3,v_5,v_2\}$, displayed on the right, which is an $\varepsilon$-approximate V-representation of $P$, but not a strong one. Although $P'$ coincides with $P$ in Figure \ref{['fig_fail']}, the algorithm does not fail in the next iteration, see Figure 3.
• Figure 3: We start a new iteration with the result of Figure \ref{['fig1a']}. $P$ is the same as $P$ in Figure \ref{['fig_fail']}. Because of the redundant inequality (brown), which is missing in Figure \ref{['fig_fail']}, Algorithm \ref{['alg_basic_cut']} does not fail here. The redundant inequality makes the line segment between $v_1$ and $v_2$ "observable" for the algorithm. Thus, in contrast to Figure \ref{['fig_fail']}, the point $v_8$ is added to $\mathcal{V}'$.
• Figure 4: The polytope $P$ from Example \ref{['ex_A2']} for $\delta = \frac{1}{10}$. For $\varepsilon=3\delta$, the set $\mathcal{V}=\{v_2,u_3,\dots,u_7\}$ provides a strong $\varepsilon$-approximate V-representation of $P$, where $v_2$ covers both $u_1$ and $u_2$. The cut with the half-space $H_\leq$ and the resulting new vertices are shown in red. The set $\mathcal{V}'=\{u_3,u_4,u_6,u_7,v_9,v_{10},u_{11},u_{12}\}$ computed by Algorithm \ref{['alg_basic_cut']} is not a strong $\varepsilon$-approximate V-representation of $P'=P\cap H$ since the vertices $u_1$ and $u_8$ of $P'$ are not covered.
• Figure 5: Schlegel diagrams of $P$ (left) and $P'$ (right) from Example \ref{['ex_A2']} showing the inequality-vertex incidence information.

Theorems & Definitions (20)

• Proposition 1
• proof
• Proposition 2
• proof
• Remark 3
• Theorem 4
• proof
• Example 5
• Theorem 6
• proof