We construct a theory of holant clones to capture the notion of expressibility in the holant framework. Their role is analogous to the role played by functional clones in the study of weighted counting Constraint Satisfaction Problems. We explore the landscape of conservative holant clones and determine the situations in which a set $\mathcal{F}$ of functions is "universal in the conservative case", which means that all functions are contained in the holant clone generated by $\mathcal{F}$ together with all unary functions. When $\mathcal{F}$ is not universal in the conservative case, we give concise generating sets for the clone. We demonstrate the usefulness of the holant clone theory by using it to give a complete complexity-theory classification for the problem of approximating the solution to conservative holant problems. We show that approximation is intractable exactly when $\mathcal{F}$ is universal in the conservative case.

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Holant克隆和保守holant问题的近似性

我们构建了一个 holant 克隆理论来捕捉 holant 框架中的可表达性概念。它们的作用类似于功能克隆在加权计数约束满足问题的研究中所起的作用。我们探索了保守 holant 克隆的景观，并确定了一组 $\mathcal{F}$ 函数在“保守情况下是通用的”的情况，这意味着所有函数都包含在 $\mathcal 生成的 holant 克隆中{F}$ 连同所有一元函数。当 $\mathcal{F}$ 在保守情况下不通用时，我们为克隆给出简明的生成集。我们通过使用 holant 克隆理论为近似保守 holant 问题的解决方案的问题提供完整的复杂性理论分类来证明其有用性。