  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.
