The integration of ontologies builds knowledge structures which brings new understanding on existing terminologies and their associations. With the steady increase in the number of ontologies, automatic integration of ontologies is preferable over manual solutions in many applications. However, available works on ontology integration are largely heuristic without guarantees on the quality of the integration results. In this work, we focus on the integration of ontologies with hierarchical

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... ures. We identified optimal structures in this problem and proposed optimal and efficient approximation algorithms for integrating a pair of ontologies. Furthermore, we extend the basic problem to address the integration of a large number of ontologies, and correspondingly we proposed an efficient approximation algorithm for integrating multiple ontologies. The empirical study on both real ontologies and synthetic data demonstrates the effectiveness of our proposed approaches. In addition, the results of integration between gene ontology and National Drug File Reference Terminology suggest that our method provides a novel way to perform association studies between biomedical terms.