Choosing a variable ordering for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition

TL;DR

This paper investigates how variable ordering affects the efficiency of a new incremental complex space CAD algorithm, evaluates existing heuristics, and proposes a new heuristic aligned with the algorithm's mechanics.

Contribution

It introduces a new heuristic for variable ordering in incremental complex space CAD, improving problem tractability based on the algorithm's mechanics.

Findings

Existing heuristics are evaluated and compared.

The new heuristic shows improved performance in certain cases.

Insights into the impact of variable ordering on CAD complexity.

Abstract

Cylindrical algebraic decomposition (CAD) is a key tool for solving problems in real algebraic geometry and beyond. In recent years a new approach has been developed, where regular chains technology is used to first build a decomposition in complex space. We consider the latest variant of this which builds the complex decomposition incrementally by polynomial and produces CADs on whose cells a sequence of formulae are truth-invariant. Like all CAD algorithms the user must provide a variable ordering which can have a profound impact on the tractability of a problem. We evaluate existing heuristics to help with the choice for this algorithm, suggest improvements and then derive a new heuristic more closely aligned with the mechanics of the new algorithm.