We present fully polynomial approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures known as polymers in statistical physics. Our method involves establishing zero-free regions for the partition functions of polymer models and using the most significant terms of the cluster expansion to approximate them. Results of our technique include new approximation and sampling algorithms for a diverse class of Holant polynomials in the low-temperature regime and approximation algorithms for general Holant problems with small signature weights. Additionally, we give randomised approximation and sampling algorithms with faster running times for more restrictive classes. Finally, we improve the known zero-free regions for a perfect matching polynomial.

中文翻译：

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复平面上 Holant 多项式的零点和近似

我们针对具有复杂边权重的一类 Holant 问题提出了完全多项式近似方案，我们将其称为 Holant 多项式。我们将这些问题转化为统计物理学中称为聚合物的抽象组合结构的分配函数。我们的方法包括为聚合物模型的分配函数建立零自由区域，并使用簇扩展的最重要项来近似它们。我们的技术的结果包括用于低温状态下各种 Holant 多项式的新近似和采样算法以及用于具有小特征权重的一般 Holant 问题的近似算法。此外，我们为更严格的类别提供了具有更快运行时间的随机近似和采样算法。最后，