Holant problems are a general framework to study counting problems. Both counting constraint satisfaction problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for \(\text {Holant}^{*}(\mathcal {F})\), where \({\mathcal {F}}\) is a set of constraint functions on Boolean variables and taking complex values. The constraint functions need not be symmetric functions. We identify four classes of problems which are polynomial time computable; all other problems are proved to be #P-hard. The main proof technique and indeed the formulation of the theorem use holographic algorithms and reductions. By considering these counting problems with the broader scope that allows complex-valued constraint functions, we discover surprising new tractable classes, which are associated with isotropic vectors, i.e., a (non-zero) vector whose dot product with itself is zero.

整体问题是研究计数问题的通用框架。计数约束满足问题（#CSP）和图同态都是特殊情况。我们证明\（\ text {Holant} ^ {*}（\ mathcal {F}）\）的复杂性二分定理，其中\（{\ mathcal {F}} \）是一组关于布尔变量并采用复杂值的约束函数。约束函数不必是对称函数。我们确定了可以用多项式时间计算的四类问题。所有其他问题都证明是#P困难的。主要的证明技术以及定理的表述都使用全息算法和归约法。通过在允许复数值约束函数的更广泛范围内考虑这些计数问题，我们发现了令人惊讶的新易处理类，这些类与各向同性矢量（即，其点积自身为零的（非零）矢量）相关联。