  The three-way table problem is to decide if there exists an l x m x n table
satisfying given line sums, and find a table if there is one. It is NP-complete
already for l=3 and every bounded integer program can be isomorphically
represented in polynomial time for some m and n as some 3 x m x n table
problem. Recently, the problem was shown to be fixed-parameter tractable with
parameters l,m. Here we extend this and show that the huge version of the
problem, where the variable side n is a huge number encoded in binary, is also
fixed-parameter tractable with parameters l,m. We also conclude that the huge
multicommodity flow problem with m suppliers and a huge number n of consumers
is fixed-parameter tractable parameterized by the numbers of commodities and
consumer types.
  One of our tools is a theorem about unimodular monoids which is of interest
on its own right. The monoid problem is to decide if a given integer vector is
a finite nonnegative integer combination of a given set of integer vectors, and
find such a decomposition if one exists. We consider sets given implicitly by
an inequality system. For such sets, it was recently shown that in fixed
dimension the problem is solvable in polynomial time with degree which is
exponential in the dimension. Here we show that when the inequality system
which defines the set is defined by a totally unimodular matrix, the monoid
problem can be solved in polynomial time even in variable dimension.
