CADS: Cylindrical Algebraic Sub-Decompositions

• CADS are focused subsets of cylindrical algebraic decompositions that isolate only the cells relevant to specific semi-algebraic problems, reducing unnecessary computation.
• Variety sub-CADs and layered sub-CADs refine full CADs by leveraging equational constraints and dimensional filtering to significantly cut computational complexity.
• The approach yields dramatic savings in cell counts and runtime, enabling efficient quantifier elimination, robot motion planning, and scalable real algebraic analysis.

Cylindrical Algebraic Sub-Decompositions (sub-CADs) extend the theory and practice of cylindrical algebraic decomposition (CAD), providing a means to focus symbolic or computational analysis on exactly those regions of ℝⁿ most relevant to a given semi-algebraic problem instance. The sub-CAD paradigm enables dramatic savings in computational complexity, cell count, and runtime for tasks in real algebraic geometry, quantifier elimination, and applications such as robot motion planning.

1. From Full CADs to Sub-CADs: Definitions and Principles

A cylindrical algebraic decomposition (CAD) of ℝⁿ, given a variable ordering $x_1 \prec x_2 \prec \cdots \prec x_n$ and a set of input polynomials $F \subset \mathbb{Q}[x_1,\dots,x_n]$, is a finite partition of ℝⁿ into semi-algebraic cells, with the two essential properties:

• Cylindricity: For any pair of cells $C$, $C'$, and any $k < n$, the projections $\pi_k(C), \pi_k(C')$ onto the first $k$ variables are either identical or disjoint.
• Sign-invariance: Each $f \in F$ has constant sign (−, 0, +) on every cell.

A sub-CAD $\mathcal{E} \subseteq \mathcal{D}$ is any subset of the cells of a given CAD $\mathcal{D}$. A sub-CAD is sign-invariant for $F \subset \mathbb{Q}[x_1,\dots,x_n]$0 if all $F \subset \mathbb{Q}[x_1,\dots,x_n]$1 are sign-invariant on every $F \subset \mathbb{Q}[x_1,\dots,x_n]$2. More generally, for a quantifier-free formula $F \subset \mathbb{Q}[x_1,\dots,x_n]$3, a sub-CAD is $F \subset \mathbb{Q}[x_1,\dots,x_n]$4-sufficient if it contains all the cells where $F \subset \mathbb{Q}[x_1,\dots,x_n]$5 holds. This framework admits focused decompositions directly aligned with the logical constraints of the problem, rather than always requiring a full decomposition of the ambient space (Wilson et al., 2014).

2. Variety and Layered Sub-CADs: Mechanisms and Algorithms

Two principal types of sub-CAD are formalized:

• Variety sub-CAD (V-sub-CAD): When a problem contains (or implies) an equational constraint $F \subset \mathbb{Q}[x_1,\dots,x_n]$6, all solutions must lie on the variety $F \subset \mathbb{Q}[x_1,\dots,x_n]$7. A V-sub-CAD is formed by extracting only those cells of a full CAD lying on $F \subset \mathbb{Q}[x_1,\dots,x_n]$8. By construction, this includes all solutions and is $F \subset \mathbb{Q}[x_1,\dots,x_n]$9-sufficient. The algorithm proceeds by (1) projecting using projection operators that exploit the constraint, (2) constructing a CAD in lower dimensions, and (3) lifting using only the equational factors, extracting only their sections (cells lying on $C$0).
• Layered sub-CAD ($C$1-L-sub-CAD): CADs naturally decompose space into cells by dimension. The union of the top $C$2 “layers” (cells of dimensions $C$3) forms an $C$4-layered sub-CAD. This is critical when only full- or high-dimensional solutions are meaningful (e.g., when seeking generic, rather than degenerate, solutions).

Combined sub-CADs—such as layered variety sub-CADs—select cells by both dimensionality and variety constraint, supporting further cell count minimization.

Algorithmic pseudocode for generating these classes of sub-CADs can be summarized as follows:

• VarietySubCAD: Project using equational-constraint operators, lift over lower-dimensional CAD, generate stacks with respect to $C$5, and extract only even-indexed (section) cells corresponding to $C$6.
• LayeredSubCAD: After projection, in each lifting stage, generate only stacks for base cells whose dimension supports the desired layer, and finally filter to desired dimensions.
• LayeredVarietySubCAD: Combine the above, applying projections with equational constraints, layered lifting, and selection of variety cells.

These constructions are compatible with Truth-Table Invariant CAD (TTICAD), allowing further logical structure exploitation (Wilson et al., 2014).

3. Complexity Analysis and Theoretical Savings

The standard CAD algorithm exhibits doubly-exponential complexity in $C$7 (number of variables), as in

$C$8

where $C$9 is the maximum total degree, $C'$0 is the number of input polynomials, and $C'$1 reflects coefficient-length complexity.

• With one equational constraint (V-sub-CAD): The constant in the exponential drops—from $C'$2 to $C'$3—by exploiting the equational constraint, resulting in substantial practical savings.
• For 1-layered variety sub-CADs: The exponent further decreases, replacing $C'$4 by $C'$5.
• Since $C'$6 grows much faster than any constant, even a single step drop in this base has a profound effect on feasibility (Wilson et al., 2014).

4. Illustrative Examples and Empirical Gains

Concrete examples illustrate the benefits:

• Bivariate case: For $C'$7, $C'$8, and formula $C'$9:
  • Full sign-invariant CAD: 23 cells.
  • Variety sub-CAD: 8 cells (only those on the unit circle $k < n$0).
  • 1-layered variety sub-CAD: 4 cells (only full-dimensional arcs on $k < n$1).
  • This realizes a 65% reduction for the V-sub-CAD, then a further 50% for the layered sub-CAD.
• ℝ³ degree-2 surfaces: The full CAD yields 17,047 cells. With an equational constraint, CAD reduces to 1,315 cells, the V-sub-CAD to 422 cells, the 1-layered variety sub-CAD to 138 cells—totaling a nearly two orders-of-magnitude reduction.
• Piano-movers (ℝ⁴): Full CAD constructs 285,419 cells in 5 hours; the 1-layered variety sub-CAD reduces this to 101,924 cells in 5 minutes (Wilson et al., 2014).

Table: Cell Count Reductions in Example Problems

Problem	Full CAD	V-sub-CAD	1-layered V-sub-CAD
Bivariate circle-with-inequality (ℝ²)	23	8	4
ℝ³ degree-2 random surfaces	17,047	422	138
Piano movers (ℝ⁴)	285,419	–	101,924

5. Maple Implementation and Practical Performance

All the aforementioned methodologies are implemented in the ProjectionCAD package for Maple, supporting experimental verification. Timings from the benchmarks confirm that sub-CAD techniques yield not only lower cell counts but also proportionally reduced runtime, making feasible analyses of high-dimensional algebraic problems that would otherwise be intractable on full CADs.

For instance, in the piano-movers configuration, reducing from hundreds of thousands to tens of thousands of cells directly translates to multi-hour to multi-minute runtime decreases (Wilson et al., 2014).

6. Broader Implications and Applications

The sub-CAD paradigm revolutionizes the practical portability of CAD-based symbolic computation:

• Quantifier elimination: Focused sub-CADs can target only the loci critical for the quantifiers under consideration.
• Robot motion planning: Layered or variety sub-CADs can reduce the configuration space to those cells representing feasible configurations, e.g., robot paths constrained to manifolds.
• Semi-algebraic geometry and topology: Working with high-dimensional varieties, layered sub-CADs facilitate the study of generic behavior, tangible topological features, and numerical tractability.
• Combined logical formulae: Integration with TTICAD permits alignment with complex logical constraints beyond equational ones.

As a result, cylindrical algebraic sub-decompositions—particularly the variety and layered sub-CADs and their TTICAD integrants—constitute a critical advance in scaling real algebraic geometry to more complex and high-dimensional regimes (Wilson et al., 2014).

7. Conclusion

Cylindrical algebraic sub-decompositions supply computational real algebraic geometry with a toolset for problem-dependent complexity mitigation. By localizing decomposition to relevant varieties and/or leveraging cell dimensionality, sub-CADs facilitate orders-of-magnitude gains in both theoretical and empirical performance. The concurrent compatibility with advanced logical decomposition, and integration into symbolic computation systems, underpins their significance for future algebraic problem solving at scale (Wilson et al., 2014).