[Paper Review] On the chordality of polynomial sets in triangular decomposition in top-down style

Chenqi Mou, Yang Bai|arXiv (Cornell University)|Feb 6, 2018

Polynomial and algebraic computation20 references3 citations

TL;DR

This paper investigates the chordal graph structures of polynomial sets during triangular decomposition in top-down style, proving that when the input polynomial set has a chordal associated graph, all intermediate polynomial sets—including triangular sets—computed by Wang’s method have associated graphs that are subgraphs of the input’s chordal graph. This structural preservation enables refined complexity analysis and potential efficiency gains for sparse polynomial systems.

ABSTRACT

In this paper the chordal graph structures of polynomial sets appearing in triangular decomposition in top-down style are studied when the input polynomial set to decompose has a chordal associated graph. In particular, we prove that the associated graph of one specific triangular set computed in any algorithm for triangular decomposition in top-down style is a subgraph of the chordal graph of the input polynomial set and that all the polynomial sets including all the computed triangular sets appearing in one specific simply-structured algorithm for triangular decomposition in top-down style (Wang's method) have associated graphs which are subgraphs of the the chordal graph of the input polynomial set. These subgraph structures in triangular decomposition in top-down style are multivariate generalization of existing results for Gaussian elimination and may lead to specialized efficient algorithms and refined complexity analyses for triangular decomposition of chordal polynomial sets.

Motivation & Objective

• To analyze the graph structures of polynomial sets arising in triangular decomposition using chordal graphs.
• To understand how chordal properties of the input polynomial set influence the decomposition process in top-down style.
• To establish theoretical guarantees on graph substructure preservation during decomposition, particularly for Wang’s method.
• To explore applications of chordal graph structures in expressing variable sparsity and refining complexity analysis of triangular decomposition.
• To extend insights from Gaussian elimination to multivariate polynomial systems via graph-theoretic methods.

Proposed method

• Define the associated graph of a polynomial set based on shared variables between polynomials.
• Introduce the concept of a chordal polynomial set as one whose associated graph is chordal.
• Prove that for any algorithm in top-down style, the associated graph of a computed triangular set is a subgraph of the input’s chordal graph.
• Reformulate Wang’s method as a decomposition tree to analyze structural invariants across all intermediate polynomial sets.
• Demonstrate that all intermediate polynomial sets in Wang’s method inherit subgraph properties from the input chordal graph.
• Propose weighted and unweighted measures of variable sparsity using associated graphs for complexity analysis.

Experimental results

Research questions

• RQ1 How do chordal graph structures of input polynomial sets affect the decomposition process in top-down triangular decomposition?
• RQ2 Under what conditions is the associated graph of a triangular set computed by a top-down algorithm a subgraph of the input’s chordal graph?
• RQ3 Can the chordal structure of the input polynomial set be preserved throughout the entire decomposition process in Wang’s method?
• RQ4 What is the relationship between the treewidth of the input graph and the complexity of triangular decomposition in top-down style?
• RQ5 How can chordal graph structures be leveraged to refine complexity analysis and improve efficiency in polynomial system solving?

Key findings

• The associated graph of any triangular set computed by an arbitrary top-down algorithm is a subgraph of the input polynomial set’s chordal associated graph.
• In Wang’s method, all intermediate polynomial sets—including all computed triangular sets—have associated graphs that are subgraphs of the input’s chordal graph.
• The associated graph of the first triangular set computed by Wang’s method is identical to that of the input polynomial set.
• The associated graphs of the other triangular sets computed by Wang’s method are strict subgraphs of the input’s chordal graph.
• Variable sparsity of polynomial sets can be quantitatively measured via the edge density of their associated graphs.
• Weighted variable sparsity, based on the number of polynomials sharing variable pairs, provides a refined measure for complexity analysis.