A kernelization for a parameterized decision problem $\mathcal{Q}$ is a polynomial-time preprocessing algorithm that reduces any parameterized instance $(x,k)$ into an instance $(x',k')$ whose size is bounded by a function of $k$ alone and which has the same yes/no answer for $\mathcal{Q}$. Such preprocessing algorithms cannot exist in the context of counting problems, when the answer to be preserved is the number of solutions, since this number can be arbitrarily large compared to $k$. However, we show that for counting minimum feedback vertex sets of size at most $k$, and for counting minimum dominating sets of size at most $k$ in a planar graph, there is a polynomial-time algorithm that either outputs the answer or reduces to an instance $(G',k')$ of size polynomial in $k$ with the same number of minimum solutions. This shows that a meaningful theory of kernelization for counting problems is possible and opens the door for future developments. Our algorithms exploit that if the number of solutions exceeds $2^{\mathsf{poly}(k)}$, the size of the input is exponential in terms of $k$ so that the running time of a parameterized counting algorithm can be bounded by $\mathsf{poly}(n)$. Otherwise, we can use gadgets that slightly increase $k$ to represent choices among $2^{O(k)}$ options by only $\mathsf{poly}(k)$ vertices.

翻译：针对参数化决策问题 $\mathcal{Q}$ 的核化是一种多项式时间预处理算法，它将任意参数化实例 $(x,k)$ 归约为一个实例 $(x',k')$，该实例的规模仅受 $k$ 的函数限制，且对 $\mathcal{Q}$ 具有相同的肯定/否定答案。在计数问题的背景下，当需要保留的答案是解的数量时，此类预处理算法无法存在，因为该数量相对于 $k$ 可能任意大。然而，我们证明，对于计数大小不超过 $k$ 的最小反馈顶点集，以及计数平面图中大小不超过 $k$ 的最小支配集，存在一种多项式时间算法：它要么直接输出答案，要么归约到一个规模为 $k$ 的多项式函数、且具有相同最小解数量的实例 $(G',k')$。这表明，为计数问题建立有意义的核化理论是可能的，并为未来的发展打开了大门。我们的算法利用如下性质：若解的数量超过 $2^{\mathsf{poly}(k)}$，则输入规模相对于 $k$ 是指数级的，从而参数化计数算法的运行时间可被界定为 $\mathsf{poly}(n)$。否则，我们可以使用部分增加 $k$ 的构造，通过仅 $\mathsf{poly}(k)$ 个顶点来表示 $2^{O(k)}$ 种选项中的选择。