Optimising Problem Formulation for Cylindrical Algebraic Decomposition

TL;DR

This paper introduces a new measure for the complexity of cylindrical algebraic decomposition (CAD) that considers the problem's real geometry, leading to improved heuristics for problem formulation and preconditioning techniques.

Contribution

It proposes a novel geometric complexity measure for CAD, enhancing heuristics for variable ordering, constraints, and problem formulation, including the use of Groebner bases.

Findings

New geometric complexity measure for CAD

Improved heuristics for variable ordering and constraints

Potential benefits of Groebner bases preconditioning

Abstract

Cylindrical algebraic decomposition (CAD) is an important tool for the study of real algebraic geometry with many applications both within mathematics and elsewhere. It is known to have doubly exponential complexity in the number of variables in the worst case, but the actual computation time can vary greatly. It is possible to offer different formulations for a given problem leading to great differences in tractability. In this paper we suggest a new measure for CAD complexity which takes into account the real geometry of the problem. This leads to new heuristics for choosing: the variable ordering for a CAD problem, a designated equational constraint, and formulations for truth-table invariant CADs (TTICADs). We then consider the possibility of using Groebner bases to precondition TTICAD and when such formulations constitute the creation of a new problem.