Choosing the Variable Ordering for Cylindrical Algebraic Decomposition via Exploiting Chordal Structure

TL;DR

This paper explores how exploiting chordal graph structures and perfect elimination orderings can optimize variable ordering in cylindrical algebraic decomposition, reducing computational complexity and resource usage.

Contribution

It introduces a novel approach linking chordal graph properties with variable ordering in CAD, demonstrating improved efficiency through perfect elimination orderings.

Findings

Perfect elimination orderings preserve chordality in CAD algorithms.

Chordal or nearly chordal graphs lead to smaller projection polynomial sets.

Lower height of the elimination tree correlates with reduced polynomial set size.

Abstract

Cylindrical algebraic decomposition (CAD) plays an important role in the field of real algebraic geometry and many other areas. As is well-known, the choice of variable ordering while computing CAD has a great effect on the time and memory use of the computation as well as the number of sample points computed. In this paper, we indicate that typical CAD algorithms, if executed with respect to a special kind of variable orderings (called "the perfect elimination orderings"), naturally preserve chordality, which is an important property on sparsity of variables. Experimentation suggests that if the associated graph of the polynomial system in question is chordal (\emph{resp.}, is nearly chordal), then a perfect elimination ordering of the associated graph (\emph{resp.}, of a minimal chordal completion of the associated graph) can be a good variable ordering for the CAD computation. That is, by using the perfect elimination orderings, the CAD computation may produce a much smaller full set of projection polynomials than by using other naive variable orderings. More importantly, for the complexity analysis of the CAD computation via a perfect elimination ordering, a so-called $(m,d)$-property of the full set of projection polynomials obtained via such an ordering is given, through which the "size" of this set is characterized. This property indicates that when the corresponding perfect elimination tree has a lower height, the full set of projection polynomials also tends to have a smaller "size". This is well consistent with the experimental results, hence the perfect elimination orderings with lower elimination tree height are further recommended to be used in the CAD projection.