Huge Unimodular N-Fold Programs

Optimization over l ×m× 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 huge version of the problem, the variable dimension n is encoded in binary, with t 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 variable. The complexity of the problem over 4-way tables with variable t remains open. Our treatment goes through the more general class of 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.