Further results on Minimal and Minimum Cylindrical Algebraic Decompositions

TL;DR

The paper investigates when a CAD adapted to a finite family of semi-algebraic sets admits a minimal or a minimum element under refinement. It develops a formal framework based on CAD reductions and an abstract reduction system, establishing that minimum existence is equivalent to system confluence and that minima in dimension $n$ are tied inductively to minima in dimension $n-1$ via projections $\\pi_{n-1}$. In low dimensions ($n=1,2$) a minimum CAD exists for every family, while in higher dimensions there are explicit counterexamples, motivating robust reduction-based algorithms (e.g., Algorithm Min) to compute minimal CADs. The work also delivers practical criteria and explicit 3D examples demonstrating both existence and non-existence of minimum CADs, highlighting how topology and connected components influence CAD optimization. Overall, the results offer a principled route to post-process CADs for minimal representation and provide concrete tools for identifying when a globally optimal CAD is attainable.

Abstract

We consider cylindrical algebraic decompositions (CADs) as a tool for representing semi-algebraic subsets of $\mathbb{R}^n$. In this framework, a CAD $\mathscr{C}$ is adapted to a given set $S$ if $S$ is a union of cells of $\mathscr{C}$. Different algorithms computing an adapted CAD may produce different outputs, usually with redundant cell divisions. In this paper we analyse the possibility to remove the superfluous data. We thus consider the set $\text{CAD}^r(\mathcal{F})$ of CADs of class $C^r$ ($r \in \mathbb{N} \cup \{\infty, ω\}$) that are adapted to a finite family $\mathcal{F}$ of semi-algebraic sets of $\mathbb{R}^n$, endowed with the refinement partial order and we study the existence of minimal and minimum element in $\text{CAD}^r(\mathcal{F})$. We show that for every such $\mathcal{F}$ and every $\mathscr{C} \in \text{CAD}^r(\mathcal{F})$, there is a minimal CAD of class $C^r$ adapted to $\mathcal{F}$ and smaller (i.e. coarser) than or equal to $\mathscr{C}$. In dimension $n=1$ or $n=2$, this result is strengthened by proving the existence of a minimum element in $\text{CAD}^r(\mathcal{F})$. In contrast, for any $n \geq 3$, we provide explicit examples of semi-algebraic sets whose associated poset of adapted CADs does not admit a minimum. We then introduce a reduction relation on $\text{CAD}^r(\mathcal{F})$ in order to define an algorithm for the computation of minimal CADs and we characterise those semi-algebraic sets $\mathcal{F}$ for which $\text{CAD}^r(\mathcal{F})$ has a minimum by means of confluence of the associated reduction system. We finally provide practical criteria for deciding if a semi-algebraic set does admit a minimum CAD and apply them to describe various concrete examples of semi-algebraic sets, along with their minimum CAD of class $C^r$.

Figures (10)

• Figure 1: The Trousers $\mathbb{T}$ and $\mathbb{T}^\omega$
• Figure 2: The CADs $\mathscr{C}$ and $\mathscr{C}'$
• Figure 3: Semi-algebraic sets with two distinct minimal CADs: $\mathcal{B}$ (left) and $\mathcal{U}$ (right)
• Figure 4: The semi-algebraic sets $\mathbb{A}$ (left) and $\Pi$ (right)
• Figure 5: The CADs $\mathscr{C}$ and $\mathscr{C}'$ and their associated CAD trees

Theorems & Definitions (75)

• Definition 2.1: see Arnonbasu2007
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Definition 2.6
• Proposition 3.1
• proof
• Remark 3.2