Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach

TL;DR

This paper introduces chordal elimination, a novel method leveraging chordal graph structures to improve the efficiency of computing Gr"obner bases for polynomial systems, especially in cases with bounded treewidth.

Contribution

The paper develops a new chordal elimination technique that exploits graph structures to enhance Gr"obner basis computations, outperforming standard methods in many scenarios.

Findings

Chordal elimination outperforms standard Gr"obner basis algorithms in many cases.

Computational complexity can be linear in the number of variables for certain ideals.

Applicable to problems in graph coloring, cryptography, sensor localization, and differential equations.

Abstract

Chordal structure and bounded treewidth allow for efficient computation in numerical linear algebra, graphical models, constraint satisfaction and many other areas. In this paper, we begin the study of how to exploit chordal structure in computational algebraic geometry, and in particular, for solving polynomial systems. The structure of a system of polynomial equations can be described in terms of a graph. By carefully exploiting the properties of this graph (in particular, its chordal completions), more efficient algorithms can be developed. To this end, we develop a new technique, which we refer to as chordal elimination, that relies on elimination theory and Gr\"obner bases. By maintaining graph structure throughout the process, chordal elimination can outperform standard Gr\"obner basis algorithms in many cases. The reason is that all computations are done on "smaller" rings, of size equal to the treewidth of the graph. In particular, for a restricted class of ideals, the computational complexity is linear in the number of variables. Chordal structure arises in many relevant applications. We demonstrate the suitability of our methods in examples from graph colorings, cryptography, sensor localization and differential equations.