Weighted counting problems are a natural generalization of counting problems where a weight is associated with every computational path of polynomial-time non-deterministic Turing machines. The goal is to compute the sum of weights of all paths (instead of number of accepting paths). Useful properties and plenty of applications make them interesting. The definition captures even undecidable problems, but obtaining an exponentially small additive approximation is just as hard as solving conventional counting. We present a structured view by defining classes that depend on the functions that assign weights to paths and by showing their relationships and how they generalize counting problems. Weighted counting is flexible and allows us to cast a number of famous results of computational complexity, including quantum computation, probabilistic graphical models and stochastic combinatorial optimization. Using the weighted counting terminology, we are able to simplify and to answer some open questions.

加权计数问题是计数问题的自然概括，其中权重与多项式时间非确定性图灵机的每个计算路径相关。目的是计算所有路径的权重之和（而不是接受路径的数量）。有用的属性和大量的应用程序使它们变得有趣。该定义甚至捕获了无法确定的问题，但是获得指数较小的加法近似值与解决常规计数一样困难。通过定义依赖于为路径分配权重的函数的类并显示它们之间的关系以及它们如何概括计数问题，我们提供了一种结构化视图。加权计数非常灵活，可以让我们转换许多著名的计算复杂性结果，包括量子计算，概率图形模型和随机组合优化。使用加权计数术语，我们可以简化并回答一些未解决的问题。