A Comparison of Three Heuristics to Choose the Variable Ordering for Cylindrical Algebraic Decomposition

Abstract

Cylindrical algebraic decomposition (CAD) is a key tool for problems in real algebraic geometry and beyond. When using CAD there is often a choice over the variable ordering to use, with some problems infeasible in one ordering but simple in another. Here we discuss a recent experiment comparing three heuristics for making this choice on thousands of examples. BackgroundA cylindrical algebraic decomposition (CAD) dissects R n into cells, each described by polynomial relations and arranged cylindrically so that the projection of any two cells into lower coordinates is equal or disjoint. CAD is a key tool, both for its original motivation of quantifier elimination (QE) over real-closed fields and many other applications discovered since. For a more detailed introduction see Huang et al. [4] and the references within. When using CAD there may be a choice for the variable ordering and it is well known that this can have a great effect on the tractability of a problem. We recently tested the following three heuristics for picking this ordering.Brown: Suggested by Brown [2], this heuristic chooses a variable ordering according to three criteria on the input system. We start with the first and break ties with successive ones. The advice is to eliminate a variable first if: 1. it has lower overall degree in the input; 2. it has lower (maximum) total degree of those terms in the input in which it occurs; 3. there is a smaller number of terms in the input which contain the variable.sotd: Suggested by Dolzmann et al. [3], this heuristic constructs the full set of projection polynomials for each ordering and selects the ordering whose set has the lowest sum of total degree for each of the monomials in each of the polynomials.ndrr: Suggested by Bradford et al.[1], this heuristic also constructs the full projection set, and selects the one with the lowest number of distinct real roots of the univariate polynomials.1