The effect of homogeneity on the computational complexity of combinatorial data anonymization

A matrix M is said to be k-anonymous if for each row r in M there are at least k − 1 other rows in M which are identical to r. The NP-hard k-Anonymity problem asks, given an n × m-matrix M over a fixed alphabet and an integer s > 0, whether M can be made k-anonymous by suppressing (blanking out) at most s entries. Complementing previous work, we introduce two new “data-driven” parameterizations for k-Anonymity—the number tin of different input rows and the number tout of different output rows—both modeling aspects of data homogeneity. We show that k-Anonymity is fixed-parameter tractable for the parameter tin, and that it is NP-hard even for tout = 2 and alphabet size four. Notably, our fixed-parameter tractability result implies that k-Anonymity can be solved in linear time when tin is a constant. Our computational hardness results also extend to the related privacy problems p-Sensitivity and ℓ-Diversity, while our fixed-parameter tractability results extend to p-Sensitivity and the usage of domain generalization hierarchies, where the entries are replaced by more general data instead of being completely suppressed.