Huge Unimodular N-Fold Programs

Optimization over $l\times m\times n$ integer $3$-way tables with given line-sums is NP-hard already for fixed $l=3$, but is polynomial time solvable with both $l,m$ fixed. In the {\em huge} version of the problem, the variable dimension $n$ is encoded in {\em binary}, with $t$ {\em layer types}. It was recently shown that the huge problem can be solved in polynomial time for fixed $t$, and the complexity of the problem for variable $t$ was raised as an open problem. Here we solve this problem and show that the huge table problem can be solved in polynomial time even when the number $t$ of types is {\em variable}. The complexity of the problem over $4$-way tables with variable $t$ remains open. Our treatment goes through the more general class of {\em huge $n$-fold integer programming problems}. We show that huge integer programs over $n$-fold products of totally unimodular matrices can be solved in polynomial time even when the number $t$ of brick types is variable.