Given a schema and a set of concepts, representative of entities in the domain of discourse, schema cover defines correspondences between concepts and parts of the schema. Schema cover aims at interpreting the schema in terms of concepts and thus, vastly simplifying the task of schema integration. In this work we investigate two properties of schema cover, namely completeness and ambiguity. The former measures the part of a schema that can be covered by a set of concepts and the latter examines the amount of overlap between concepts in a cover. To study the tradeoffs between completeness and ambiguity we define a cover model to which previous frameworks are special cases. We analyze the theoretical complexity of variations of the cover problem, some aim at maximizing completeness while others aim at minimizing ambiguity. We show that variants of the schema cover problem are hard problems in general and formulate an exhaustive search solution using integer linear programming. We then provide a thorough empirical analysis, using both real-world and simulated data sets, showing empirically that the integer linear programming solution scales well for large schemata. We also show that some instantiations of the general schema cover problem are more effective in identifying correct attribute correspondences than others.