Before revision:  To this end, given \to  an integer k, we devise algorithms that approximately enumerate all minimal subsets of U with cardinality at most k satisfying \Pi for various monotone properties \Pi, where "approximate enumeration" means that algorithms  may output some minimal subsets satisfying \Pi whose cardinality exceeds k and is at most ck for some constant c \ge 1.
After revision:  To this end, given a weight function w: U\to an integer k, we devise algorithms that approximately enumerate all minimal subsets of U with cardinality at most k satisfying \Pi for various monotone properties \Pi, where "approximate enumeration" means that algorithms may output some minimal subsets satisfying \Pi whose cardinality exceeds k and is at most ck for some constant c \ge 1.