In this paper, we introduce and study the Minimum Consistent Subset Cover (MCSC) problem. Given a finite ground set X and a constraint t, find the minimum number of consistent subsets that cover X, where a subset of X is consistent if it satisfies t. The MCSC problem generalizes the traditional set covering problem and has Minimum Clique Partition, a dual problem of graph coloring, as an instance. Many practical data mining problems in the areas of rule learning, clustering, and frequent pattern mining can be formulated as MCSC instances. In particular, we discuss the Minimum Rule Set problem that minimizes model complexity of decision rules as well as some converse k-clustering problems that minimize the number of clusters satisfying certain distance constraints. We also show how the MCSC problem can find applications in frequent pattern summarization. For any of these MCSC formulations, our proposed novel graph-based generic algorithm CAG can be directly applicable. CAG starts by constructing a maximal optimal partial solution, then performs an example-driven specific-to-general search on a dynamically maintained bipartite assignment graph to simultaneously learn a set of consistent subsets with small cardinality covering the ground set. Our experiments on benchmark datasets show that CAG achieves good results compared to existing popular heuristics.