A property $\Pi$ on a finite set $U$ is \emph{monotone} if for every $X \subseteq U$ satisfying $\Pi$, every superset $Y \subseteq U$ of $X$ also satisfies $\Pi$. Many combinatorial properties can be seen as monotone properties. The problem of finding a minimum subset of $U$ satisfying $\Pi$ is a central problem in combinatorial optimization. Although many approximate/exact algorithms have been developed to solve this kind of problem on numerous properties, a solution obtained by these algorithms is often unsuitable for real-world applications due to the difficulty of building accurate mathematical models on real-world problems. A promising approach to overcome this difficulty is to \emph{enumerate} multiple small solutions rather than to \emph{find} a single small solution. To this end, given a weight function $w: U \to \mathbb N$ and an integer $k$, we devise algorithms that \emph{approximately} enumerate all minimal subsets of $U$ with weight at most $k$ satisfying $\Pi$ for various monotone properties $\Pi$, where "approximate enumeration" means that algorithms output all minimal subsets satisfying $\Pi$ whose weight at most $k$ and may output some minimal subsets satisfying $\Pi$ whose weight exceeds $k$ but is at most $ck$ for some constant $c \ge 1$. These algorithms allow us to efficiently enumerate minimal vertex covers, minimal dominating sets in bounded degree graphs, minimal feedback vertex sets, minimal hitting sets in bounded rank hypergraphs, etc., of weight at most $k$ with constant approximation factors.

翻译：若对于有限集合$U$上满足属性$\Pi$的任意子集$X \subseteq U$，其任意超集$Y \subseteq U$也满足$\Pi$，则称该属性是\emph{单调的}。许多组合属性均可视为单调属性。在组合优化中，寻找满足$\Pi$的$U$的最小子集是一个核心问题。尽管针对众多属性已开发出许多近似/精确算法来解决此类问题，但由于现实问题难以建立精确数学模型，这些算法得到的解通常不适用于实际应用。克服此困难的一种有效方法是通过\emph{枚举}多个较小解而非\emph{寻找}单个小解。为此，给定权重函数$w: U \to \mathbb N$与整数$k$，我们针对各类单调属性$\Pi$设计了能够\emph{近似}枚举所有权重不超过$k$且满足$\Pi$的$U$最小子集的算法，其中“近似枚举”指算法输出所有权重不超过$k$且满足$\Pi$的最小子集，同时可能输出部分满足$\Pi$但权重超过$k$（不超过$ck$，其中$c \ge 1$为常数）的最小子集。这些算法使我们能够以常数近似因子，高效枚举有界度图中的最小顶点覆盖、最小支配集、最小反馈顶点集，以及有界秩超图中的最小命中集等权重不超过$k$的结构。