# Comparing machine learning models to choose the variable ordering for cylindrical algebraic decomposition

## TL;DR

This paper explores machine learning techniques to directly select variable orderings for cylindrical algebraic decomposition, outperforming traditional heuristics in computational efficiency on specific datasets.

## Contribution

It extends prior work by enabling ML to choose variable orderings directly and evaluates multiple ML models against established heuristics.

## Key findings

- ML models outperform human heuristics in selecting variable orderings
- ML approaches significantly reduce computation time
- Multiple ML techniques show comparable or superior performance

## Abstract

There has been recent interest in the use of machine learning (ML) approaches within mathematical software to make choices that impact on the computing performance without affecting the mathematical correctness of the result. We address the problem of selecting the variable ordering for cylindrical algebraic decomposition (CAD), an important algorithm in Symbolic Computation. Prior work to apply ML on this problem implemented a Support Vector Machine (SVM) to select between three existing human-made heuristics, which did better than anyone heuristic alone. The present work extends to have ML select the variable ordering directly, and to try a wider variety of ML techniques. We experimented with the NLSAT dataset and the Regular Chains Library CAD function for Maple 2018. For each problem, the variable ordering leading to the shortest computing time was selected as the target class for ML. Features were generated from the polynomial input and used to train the following ML models: k-nearest neighbours (KNN) classifier, multi-layer perceptron (MLP), decision tree (DT) and SVM, as implemented in the Python scikit-learn package. We also compared these with the two leading human constructed heuristics for the problem: Brown's heuristic and sotd. On this dataset all of the ML approaches outperformed the human made heuristics, some by a large margin.