Bounds on Entries in 3-Dimensional Contingency Tables Subject to Given Marginal Totals

Problems in statistical data security have led to interest in determining exact integer bounds on entries in multi-dimensional contingency tables subject to fixed marginal totals. We investigate the 3-dimensional integer planar transportation problem (3-DIPTP). Heuristic algorithms for bounding entries in 3-DIPTPs have recently appeared. We demonstrate these algorithms are not exact, are based on necessary but not sufficient conditions to solve 3-DIPTP, and that all are insensitive to whether a feasible table exists. We compare the algorithms and demonstrate that one is superior, but not original. We exhibit fractional extremal points and discuss implications for statistical data base query systems.