利用弦结构选择柱形代数分解的变量序
Choosing the Variable Ordering for Cylindrical Algebraic Decomposition via Exploiting Chordal Structure
摘要：
柱形代数分解（CAD）在实代数几何等领域有着重要的作用。众所周知，计算CAD时变量排序的选择对计算的时间和内存使用以及计算的样本点数量有很大影响。在本文中，我们指出，典型的CAD算法，如果执行一种特殊的变量排序（称为“完美消除排序”），自然保持弦，这是一个重要的属性稀疏的变量。实验表明，如果所讨论的多项式系统的关联图是弦的（\emph{resp.}，几乎是弦的），则关联图的完美消除排序（\emph{resp.}，关联图的最小弦完成）可以是用于CAD计算的良好变量排序。也就是说，通过使用完美消除排序，CAD计算可以产生比通过使用其他朴素变量排序小得多的投影多项式的全集。更重要的是，对于CAD计算的复杂性分析通过一个完美的消除顺序，所谓的$(m,d)$ -性质的投影多项式的完整的集合通过这样的顺序获得，通过该集合的“大小”的特点。该性质表明，当对应的完全消除树具有较低的高度时，投影多项式的全集也倾向于具有较小的“大小”。这与实验结果是一致的，因此，具有较低的消除树高度的完美消除顺序进一步推荐用于CAD投影。
英文摘要：
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.
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