Complementary cell suppression is a method for protecting data pertaining to individual respondents from statistical disclosure when the data are presented in tabular form. Several mathematical methods for complementary suppression have been proposed in the statistical literature; some have been implemented in large-scale data processing environments by national statistical agencies. Each method has either theoretical or computational limitations. This article presents solutions to the complementary cell suppression problem based on linear optimization over a mathematical network. These methods are shown to be optimal for certain problems and to offer theoretical and practical advantages, including comprehensiveness, comprehensibleness, and computational efficiency.