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P-Time Algorithms for Typical #EO Problems

Authors Boning Meng , Juqiu Wang , Mingji Xia

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  • Theory of computation → Problems, reductions and completeness
Keywords
  • Counting complexity
  • Eulerian orientation
  • Holant
  • #P-hardness
  • Dichotomy theorem

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Abstract

In this article, we study the computational complexity of counting weighted Eulerian orientations, denoted as #EO. This problem is considered a pivotal scenario in the complexity classification for Holant, a counting framework of great significance. Our results consist of three parts. First, we prove a complexity dichotomy theorem for #EO defined by a set of binary and quaternary signatures, which generalizes the previous dichotomy for the six-vertex model. Second, we prove a dichotomy for #EO defined by a set of so-called pure signatures, which possess the closure property under gadget construction. Finally, we present a polynomial-time algorithm for #EO defined by specific rebalancing signatures, which extends the algorithm for pure signatures to a broader range of problems, including #EO defined by non-pure signatures such as f_40. We also construct a signature f_56 that is not rebalancing, and whether #EO(f_56) is computable in polynomial time remains open.

Cite As Get BibTex

Boning Meng, Juqiu Wang, and Mingji Xia. P-Time Algorithms for Typical #EO Problems. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 118:1-118:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.118

Author Details

Boning Meng
  • Key Laboratory of System Software (Chinese Academy of Sciences) and State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China
Juqiu Wang
  • Key Laboratory of System Software (Chinese Academy of Sciences) and State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China
Mingji Xia
  • Key Laboratory of System Software (Chinese Academy of Sciences) and State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China

Funding

All authors are supported by National Key R&D Program of China (2023YFA1009500), NSFC 61932002 and NSFC 62272448.

Acknowledgements

We sincerely thank Yicheng Pan for providing a simplified proof of a lemma in the full version.

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