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

Abstract

Cylindrical algebraic decomposition (CAD) plays an important role in field of real algebraic geometry and many other areas. As is well-known, choice of variable ordering while computing CAD has a great effect on time and memory use of computation as well as 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 well compatible with an important (variable) sparsity pattern called the correlative sparsity''. Experimentation suggests that if associated graph of polynomial system in question is chordal (resp., is nearly chordal), then a perfect elimination ordering of associated graph (resp., of a minimal chordal completion of associated graph) can be a good variable ordering for CAD computation. That is, by using perfect elimination orderings, CAD computation may produce a much smaller full set of projection polynomials than by using other naive variable orderings. More importantly, for complexity analysis of CAD computation via a perfect elimination ordering, an (m,d)-property of full set of projection polynomials obtained via such an ordering is given, through which size'' of this set is characterized. This property indicates that when corresponding perfect elimination tree has a lower height, full set of projection polynomials also tends to have a smaller size''. This is well consistent with experimental results, hence perfect elimination orderings with lower elimination tree height are further recommended to be used in CAD projection.