Huge multiway table problems

Optimization over l×m×nl×m×n integer threeway tables is NP-hard already for fixed l=3l=3, but solvable in polynomial time with both l,ml,m fixed. Here we consider huge   tables, where the variable dimension nn is encoded in binary  . Combining recent results on Graver bases and recent results on integer cones, we show how to handle such problems in polynomial time. We also show that a harder variant of the problem lies in both NP and coNP. Our treatment goes through the more general class of nn-fold integer programming problems.