On the Complexity of #CSP^d

Author Jiabao Lin

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Jiabao Lin
  • Shanghai University of Finance and Economics, China


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

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Jiabao Lin. On the Complexity of #CSP^d. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 40:1-40:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Counting CSP^d 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 #CSP^d 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 tensors distinguished by these algebraic operations may share the same closure property under tensor product and contraction.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Constraint satisfaction problem
  • counting problems
  • Holant
  • complexity dichotomy


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