All Linear and Integer Programs Are Slim 3-Way Transportation Programs

We show that any rational convex polytope is polynomial-time representable as a 3-way line-sum transportation polytope of “slim” $(r,c,3)$ format. This universality theorem has important consequences for linear and integer programming and for confidential statistical data disclosure. We provide a polynomial-time embedding of arbitrary linear programs and integer programs in such slim transportation programs and in bitransportation programs. Our construction resolves several standing problems on $3$-way transportation polytopes. For example, it demonstrates that, unlike the case of $2$-way contingency tables, the range of values an entry can attain in any slim $3$-way contingency table with specified $2$-margins can contain arbitrary gaps. Our smallest such example has format $(6,4,3)$. Our construction provides a powerful automatic tool for studying concrete questions about transportation polytopes and contingency tables. For example, it automatically provides new proofs for some classical results, including a well-known “real-feasible but integer-infeasible” $(6,4,3)$-transportation polytope of M. Vlach, and bitransportation programs where any feasible bitransportation must have an arbitrarily large prescribed denominator.