Using Machine Learning to Decide When to Precondition Cylindrical Algebraic Decomposition With Groebner Bases

TL;DR

This paper explores using machine learning, specifically SVMs, to predict when preconditioning with Groebner Bases improves the efficiency of Cylindrical Algebraic Decomposition in computational algebraic geometry.

Contribution

It introduces a machine learning approach to decide when to apply Groebner Basis preconditioning for CAD, outperforming previous heuristics.

Findings

ML-based decision outperforms human heuristics

Over 1000 problem instances tested

Supports automated optimization of CAD preprocessing

Abstract

Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. However, it can be expensive, with worst case complexity doubly exponential in the size of the input. Hence it is important to formulate the problem in the best manner for the CAD algorithm. One possibility is to precondition the input polynomials using Groebner Basis (GB) theory. Previous experiments have shown that while this can often be very beneficial to the CAD algorithm, for some problems it can significantly worsen the CAD performance. In the present paper we investigate whether machine learning, specifically a support vector machine (SVM), may be used to identify those CAD problems which benefit from GB preconditioning. We run experiments with over 1000 problems (many times larger than previous studies) and find that the machine learned choice does better than the human-made heuristic.