The Complexity of Three-Way Statistical Tables

Multiway 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 outcome of our work is that each of the following problems is intractable already for “slim” 3tables, with constant number 3 of rows: (1) deciding existence of 3-tables with specified 2-marginals; (2) counting all 3-tables with specified 2-marginals; (3) deciding whether a specified value is attained in a specified entry by at least one of the 3-tables having the same 2-marginals as a given table. This implies that a characterization of feasible marginals for such slim tables, sought by much recent research, is unlikely to exist. Another 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. Remarkably, it enables us to automatically recover a famous example due to Vlach of a “real-feasible integer-infeasible” collection of 2-marginals for 3-tables of smallest possible size (3, 4, 6).