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From an Odd Arity Signature to a Holant Dichotomy

Authors Boning Meng , Juqiu Wang , Mingji Xia , Jiayi Zheng

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  • Theory of computation → Problems, reductions and completeness
Keywords
  • Complexity dichotomy
  • Counting
  • Holant problem
  • #P

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Abstract

Holant is an essential framework in the field of counting complexity. For over fifteen years, researchers have been clarifying the complexity classification for complex-valued Holant on Boolean domain, a challenge that remains unresolved. In this article, we prove a complexity dichotomy for complex-valued Holant on Boolean domain when a non-trivial signature of odd arity exists. This dichotomy is based on the dichotomy for #EO, and consequently is an FP^NP vs. #P dichotomy as well, stating that each problem is either in FP^NP or #P-hard. 
Furthermore, we establish a generalized version of the decomposition lemma for complex-valued Holant on Boolean domain. It asserts that each signature can be derived from its tensor product with other signatures, or conversely, the problem itself is in FP^NP. We believe that this result is a powerful method for building reductions in complex-valued Holant, as it is also employed as a pivotal technique in the proof of the aforementioned dichotomy in this article.

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Boning Meng, Juqiu Wang, Mingji Xia, and Jiayi Zheng. From an Odd Arity Signature to a Holant Dichotomy. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.23

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
Jiayi Zheng
  • 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) and NSFC 62272448.

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