All Rational Polytopes Are Transportation Polytopes and All Polytopal Integer Sets Are Contingency Tables

We show that any rational polytope is polynomial-time representable as a “slim” r× c × 3 three-way line-sum transportation polytope. This universality theorem has important consequences for linear and integer programming and for confidential statistical data disclosure.It provides polynomial-time embedding of arbitrary linear programs and integer programs in such slim transportation programs and in bipartite biflow programs. It resolves several standing problems on 3-way transportation polytopes. It demonstrates that the range of values an entry can attain in any slim 3-way contingency table with specified 2-margins can contain arbitrary gaps, suggesting that disclosure of k-margins of d-tables for 2≤ k<d is confidential.Our construction also provides a powerful tool in studying concrete questions about transportation polytopes and contingency tables; remarkably, it enables to automatically recover the famous “real-feasible integer-infeasible” 6× 4× 3 transportation polytope of M. Vlach, and to produce the first example of 2-margins for 6× 4× 3 contingency tables where the range of values a specified entry can attain has a gap.