An Intuitive Formulation and Solution of the Exact Cell-Bounding Problem for Contingency Tables of Conditional Frequencies

Recent research has raised questions about the structure and calculation of bounds on the underlying cell counts for a contingency table released in the form of conditional probabilities. This problem has implications for statistical disclosure limitation. We elucidate the mathematical structure of the problem in fairly elementary terms, under the assumption that the unrounded conditionals and sample size are known. To do so, we reformulate a standard integer programming approach as a knapsack problem, show that this provides many insights into the problem, and provide illustrations in the context of several datasets. In particular, we demonstrate that the tightest bounds are much easier to calculate than previously believed, and we also identify circumstances in which disclosure is either guaranteed or unlikely to occur.