New heuristic to choose a cylindrical algebraic decomposition variable ordering motivated by complexity analysis

TL;DR

This paper introduces new heuristics inspired by complexity analysis to improve variable ordering in cylindrical algebraic decomposition, leading to more efficient computations on benchmark problems.

Contribution

It proposes novel heuristics for variable ordering in CAD based on complexity analysis, outperforming existing heuristics in benchmark evaluations.

Findings

New heuristics reduce average computation time by 17% compared to the virtual-best.

Proposed metrics effectively evaluate variable ordering strategies.

Heuristics outperform prior methods on SMT-LIB benchmarks.

Abstract

It is well known that the variable ordering can be critical to the efficiency or even tractability of the cylindrical algebraic decomposition (CAD) algorithm. We propose new heuristics inspired by complexity analysis of CAD to choose the variable ordering. These heuristics are evaluated against existing heuristics with experiments on the SMT-LIB benchmarks using both existing performance metrics and a new metric we propose for the problem at hand. The best of these new heuristics chooses orderings that lead to timings on average 17% slower than the virtual-best: an improvement compared to the prior state-of-the-art which achieved timings 25% slower.