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

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Part of the Lecture Notes in Computer Science book series (LNCS,volume 13366)

Abstract

When 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.

Keywords

  • Triangular decomposition
  • Chordal graph
  • Sparsity
  • Implementation

This work was partially supported by the National Natural Science Foundation of China (NSFC 11971050) and Beijing Natural Science Foundation (Z180005).

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Notes

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    https://odebase.cs.uni-bonn.de/ODEModelApp.

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Acknowledgments

The authors would like to thank Prof. Dongming Wang for his insightful comments on the implementations in Epsilon package and the referees for their helpful comments resulting in improvements on the previous version of this paper.

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Correspondence to Chenqi Mou .

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Dong, M., Mou, C. (2022). Analyses and Implementations of Chordality-Preserving Top-Down Algorithms for Triangular Decomposition. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2022. Lecture Notes in Computer Science, vol 13366. Springer, Cham. https://doi.org/10.1007/978-3-031-14788-3_8

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  • DOI: https://doi.org/10.1007/978-3-031-14788-3_8

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