We study the complexity of $\mathrm{\#}{\mathsf{CSP}}_{\mathrm{\Delta }}(\mathrm{\Gamma })$, which is the problem of counting satisfying assignments to CSP instances with constraints from Γ and whose variables can appear at most Δ times. Our main result shows that: (i) if every function in Γ is affine, then $\mathrm{\#}{\mathsf{CSP}}_{\mathrm{\Delta }}(\mathrm{\Gamma })$ is in FP for all Δ, (ii) otherwise, if every function in Γ is in a class called $I{M}_2$, then for large Δ, $\mathrm{\#}{\mathsf{CSP}}_{\mathrm{\Delta }}(\mathrm{\Gamma })$ is equivalent under approximation-preserving reductions to the problem of counting independent sets in bipartite graphs, (iii) otherwise, for large Δ, it is $\mathsf{NP}$-hard to approximate $\mathrm{\#}{\mathsf{CSP}}_{\mathrm{\Delta }}(\mathrm{\Gamma })$, even within an exponential factor.

中文翻译：

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有界布尔运算约束满足问题的近似分区函数

我们研究了 $\mathrm{＃}{\mathsf{CSP}}_{\mathrm{\Delta }}（\mathrm{\Gamma }）$，这是对具有Γ约束且其变量最多可以出现Δ次的CSP实例进行满意分配计数的问题。我们的主要结果表明：（i）如果Γ中的每个函数都是仿射的，则$\mathrm{＃}{\mathsf{CSP}}_{\mathrm{\Delta }}（\mathrm{\Gamma }）$ 对于所有Δ都在FP中，（ii）否则，如果Γ中的每个函数都在称为 $\mathrm{一世}{\mathrm{中号}}_2$，那么对于大Δ， $\mathrm{＃}{\mathsf{CSP}}_{\mathrm{\Delta }}（\mathrm{\Gamma }）$ 在保留逼近约简下等于对二部图中的独立集合进行计数的问题，（iii）否则，对于大Δ，它是 $\mathsf{NP}$-很难估计 $\mathrm{＃}{\mathsf{CSP}}_{\mathrm{\Delta }}（\mathrm{\Gamma }）$，即使是在指数因子内。