Analyses and Implementations of Chordality-Preserving Top-Down Algorithms for Triangular Decomposition

Abstract

AbstractWhen the input polynomial set has a chordal associated graph, top-down algorithms for triangular decomposition are proved to preserve the chordal structure. Based on these theoretical results, sparse algorithms for triangular decomposition were proposed and demonstrated with experiments to be more efficient in case of sparse polynomial sets. However, existing implementations of top-down triangular decomposition are not guaranteed to be chordality-preserving due to operations which potentially destroy the chordality. In this paper, we first analyze the current implementations of typical top-down algorithms for triangular decomposition in the Epsilon package to identify these chordality-destroying operations. Then modifications are made accordingly to guarantee new implementations of such algorithms are chordality-preserving. In particular, the technique of dynamic checking is introduced to ensure that the modifications also keep the computational efficiency. Experimental results with polynomial sets from biological systems are also reported.KeywordsTriangular decompositionChordal graphSparsityImplementation