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The Complexity of Counting CSPd

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Abstract

Counting CSPd is the counting constraint satisfaction problem (# CSP in short) restricted to the instances where every variable occurs a multiple of d times. This paper revisits tractable structures in # CSP and gives a complexity classification theorem for # CSPd with algebraic complex weights. The result unifies affine functions (stabilizer states in quantum information theory) and related variants such as the local affine functions, the discovery of which leads to all the recent progress on the complexity of Holant problems. The Holant is a framework that generalizes counting CSP. In the literature on Holant problems, weighted constraints are often expressed as tensors (vectors) such that projections and linear transformations help analyze the structure. This paper gives an example showing that different classes of constraints distinguished by these algebraic operations may share the same closure property.

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Notes

  1. In [4], a stronger condition called the Block Orthogonality is imposed on the set \(\mathcal {W}_{\mathcal {F}}\), but the algorithmic part of dichotomy only requires orthogonality. As we will see in Section 2, violation of the Orthogonality condition implies # P-hardness, which is proved by using the notion of block orthogonality.

  2. For technical reasons, we require finite constraint languages in this paper. In fact, the equivalence also holds if we include equality functions of all arities.

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Acknowledgements

I would like to thank anonymous referees for their valuable comments and suggestions.

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  1. Shanghai University of Finance and Economics, Shanghai, China

    Jiabao Lin

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A preliminary version of this article appeared in the 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). This work is supported by Science and Technology Innovation 2030 –“New Generation of Artificial Intelligence” Major Project No.(2018AAA0100903), NSFC grant 61932002, Program for Innovative Research Team of Shanghai University of Finance and Economics (IRTSHUFE) and the Fundamental Research Funds for the Central Universities.

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Lin, J. The Complexity of Counting CSPd. Theory Comput Syst 66, 309–321 (2022). https://doi.org/10.1007/s00224-021-10060-x

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