On the Complexity of t-Closeness Anonymization and Related Problems

Abstract

An important issue in releasing individual data is to protect the sensitive information from being leaked and maliciously utilized. Famous privacy preserving principles that aim to ensure both data privacy and data integrity, such as k-anonymity and l-diversity, have been extensively studied both theoretically and empirically. Nonetheless, these widely-adopted principles are still insufficient to prevent attribute disclosure if the attacker has partial knowledge about the overall sensitive data distribution. The t-closeness principle has been proposed to fix this, which also has the benefit of supporting numerical sensitive attributes. However, in contrast to k-anonymity and l-diversity, the theoretical aspect of t-closeness has not yet been well investigated.

We initiate the first systematic theoretical study on the t-closeness principle under the commonly-used attribute suppression model. We prove that for every constant t such that 0 ≤ t < 1, it is NP-hard to find an optimal t-closeness generalization of a given table. The proof consists of several reductions each of which works for different values of t, which together cover the full range. To complement this negative result, we also provide exact and fixed-parameter algorithms. Finally, we answer some open questions regarding the complexity of k-anonymity and l-diversity left in the literature.

This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61033001, 61061130540, 61073174. The research of the second author was supported by the Research Grants Council of Hong Kong under grant 9041688 (CityU 124411).