The Complexity of Three-Way Statistical Tables
<|reference_start|>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.<|reference_end|>
@article{de loera2002the,
    title={The Complexity of Three-Way Statistical Tables},
    author={Jesus De Loera, Shmuel Onn},
    journal={SIAM Journal on Computing, 33:819--836, 2004},
    year={2002},
    archivePrefix={arXiv},
    eprint={math/0207200},
    primaryClass={math.CO cs.DM math.OC}
}