In recent years, concerns of privacy have become more prominent for social networks. Anonymizing a graph meaningfully is a challenging problem, as the original graph properties must be preserved as well as possible. We introduce a generalization of the degree anonymization problem posed by Liu and Terzi. In this problem, our goal is to anonymize a given subset of vertices in a graph while adding the fewest possible number of edges. We examine different approaches to solving the problem, one of which finds a degree-constrained subgraph to determine which edges to add within the given subset and another that uses a greedy approach that is not optimal, but is more efficient in space and time. The main contribution of this thesis is an efficient algorithm for this problem by exploring its connection with the degree-constrained subgraph problem. Our experimental results show that our algorithms perform very well on many instances of social network data. / Graduate