We introduce a new framework for the analysis of preprocessing routines for parameterized counting problems. Existing frameworks that encapsulate parameterized counting problems permit the usage of exponential (rather than polynomial) time either explicitly or by implicitly reducing the counting problems to enumeration problems. Thus, our framework is the only one in the spirit of classic kernelization (as well as lossy kernelization). Specifically, we define a compression of a counting problem $P$ into a counting problem $Q$ as a pair of polynomial-time procedures: $\mathsf{reduce}$ and $\mathsf{lift}$. Given an instance of $P$, $\mathsf{reduce}$ outputs an instance of $Q$ whose size is bounded by a function $f$ of the parameter, and given the number of solutions to the instance of $Q$, $\mathsf{lift}$ outputs the number of solutions to the instance of $P$. When $P=Q$, compression is termed kernelization, and when $f$ is polynomial, compression is termed polynomial compression. Our technical (and other conceptual) contributions concern both upper bounds and lower bounds.

翻译：我们提出了一种新的框架，用于分析参数化计数问题的预处理例程。现有的针对参数化计数问题的框架要么明确允许使用指数时间（而非多项式时间），要么通过隐式地将计数问题简化为枚举问题来实现这一点。因此，我们的框架是唯一一个遵循经典核化（以及有损核化）精神的框架。具体而言，我们将计数问题$P$到计数问题$Q$的压缩定义为一对多项式时间过程：$\mathsf{reduce}$和$\mathsf{lift}$。给定$P$的一个实例，$\mathsf{reduce}$输出一个$Q$的实例，其大小受参数函数$f$的限制；而给定$Q$实例的解的个数，$\mathsf{lift}$输出$P$实例的解的个数。当$P=Q$时，压缩被称为核化；当$f$是多项式时，压缩被称为多项式压缩。我们的技术性（及其他概念性）贡献涉及上界和下界。