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Dichotomy for Holant∗ Problems on the Boolean Domain

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Abstract

Holant problems are a general framework to study counting problems. Both counting constraint satisfaction problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for \(\text {Holant}^{*}(\mathcal {F})\), where \({\mathcal {F}}\) is a set of constraint functions on Boolean variables and taking complex values. The constraint functions need not be symmetric functions. We identify four classes of problems which are polynomial time computable; all other problems are proved to be #P-hard. The main proof technique and indeed the formulation of the theorem use holographic algorithms and reductions. By considering these counting problems with the broader scope that allows complex-valued constraint functions, we discover surprising new tractable classes, which are associated with isotropic vectors, i.e., a (non-zero) vector whose dot product with itself is zero.

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Notes

  1. To avoid any difficulties with models of computation, we restrict to functions taking algebraic numbers in \(\mathbb {C}\).

  2. B. Szegedy [45] studied an edge coloring model, which is identical to Holant problems on a general domain, with a single symmetric constraint function per each arity.

  3. In this paper, we actually present it slightly differently, in order to give a more succinct proof.

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Acknowledgements

We sincerely thank Miriam Backens, Xi Chen, Martin Dyer, Leslie Ann Goldberg, Zhiguo Fu, Heng Guo, Michael Kowalczyk, Les Valiant and Tyson Williams for discussions and comments. We also thank the anonymous referees of SODA 2011 who read our submission and made very constructive comments. Last but not least we sincerely thank the anonymous referees for the journal submission version. Many comments have been incorporated in this revised version with hopefully an improved presentation.

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Authors and Affiliations

  1. Computer Sciences Department, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Jin-Yi Cai

  2. School of Information Management and Engineering, Shanghai University of Finance and Economics, Yangpu District, Shanghai, China

    Pinyan Lu

  3. Stake Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Beijing, China

    Mingji Xia

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  1. Jin-Yi Cai
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Correspondence to Jin-Yi Cai.

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A preliminary version of this paper appeared in ACM-SIAM Symposium on Discrete Algorithms (SODA) 2011 [12]. Jin-Yi Cai is supported by NSF CCF-1714275. Pinyan Lu is supported by Science and Technology Innovation 2030 “New Generation of Artificial Intelligence” Major Project No.(2018AAA0100903), NSFC grant 61922052 and 61932002, Program for Innovative Research Team of Shanghai University of Finance and Economics (IRTSHUFE) and the Fundamental Research Funds for the Central Universities. Mingji Xia is supported by NSFC grant 61932002, and the Youth Innovation Promotion Association, CAS.

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Cai, JY., Lu, P. & Xia, M. Dichotomy for Holant∗ Problems on the Boolean Domain. Theory Comput Syst 64, 1362–1391 (2020). https://doi.org/10.1007/s00224-020-09983-8

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