  Multi-way tables with specified marginals arise in a variety of applications
in statistics and operations research. We provide a comprehensive complexity
classification of three fundamental computational problems on tables:
existence, counting and entry-security.
  One major outcome of our work is that each of the following problems is
intractable already for "slim" 3-tables, with constant and smallest possible
number 3 of rows: (1) deciding existence of 3-tables with given consistent
2-marginals; (2) counting all 3-tables with given 2-marginals; (3) finding
whether an integer value is attained in entry (i,j,k) by at least one of the
3-tables satisfying given (feasible) 2-marginals. This implies that a
characterization of feasible marginals for such slim tables, sought by much
recent research, is unlikely to exist.
  Another important consequence of our study is a systematic efficient way of
embedding the set of 3-tables satisfying any given 1-marginals and entry upper
bounds in a set of slim 3-tables satisfying suitable 2-marginals with no entry
bounds. This provides a valuable tool for studying multi-index transportation
problems and multi-index transportation polytopes.
