The Complexity of Three-Way Statistical Tables
  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. 
Jesus De Loera, Shmuel Onn