The Complexity of the Counting Constraint Satisfaction Problem

The Counting Constraint Satisfaction Problem (${\rm \#CSP}(\mathcal{H})$) over a finite relational structure $\mathcal{H}$ can be expressed as follows: given a relational structure $\mathcal{G}$ over the same vocabulary, determine the number of homomorphisms from $\mathcal{G}$ to $\mathcal{H}$. In this paper we characterize relational structures $\mathcal{H}$ for which ${\rm \#CSP}(\mathcal{H})$ can be solved in polynomial time and prove that for all other structures the problem is #P-complete.

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