A property \Pi on a finite set U is 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 , and 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  problem on numerous properties, a solution obtained by these algorithms is often unsuitable for real-world applications due to the difficulty of building  mathematical models on real-world problems. A promising approach to overcome this difficulty is to enumerate multiple small solutions rather than to find a single small solution. 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. 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.,  with constant approximation factors.
A property \Pi on a finite set U is 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 enumerate multiple small solutions rather than to 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 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.
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