Markov bases of three-way tables are arbitrarily complicated

We show the following two universality statements on the entry-ranges and Markov bases of spaces of 3-way contingency tables with fixed 2-margins: (1) For any finite set D of nonnegative integers, there are r,c, and 2-margins for (r,c,3)-tables such that the set of values occurring in a fixed entry in all possible tables with these margins is D. (2) For any integer n-vector d, there are r,c such that any Markov basis for (r,c,3)-tables with fixed 2-margins must contain an element whose restriction to some n entries is d. In particular, the degree and support of elements in the minimal Markov bases when r and c vary can be arbitrarily large, in striking contrast with the case for 1-margined tables in any dimension and any format and with 2-margined (r,c,h)-tables with both c,h fixed. These results have implications for confidential statistical data disclosure control. Specifically, they demonstrate that the entry-range of 2-margined 3-tables can contain arbitrary gaps, suggesting that even if the smallest and largest possible values of an entry are far apart, the disclosure of such margins may be insecure. Thus, the behavior of sensitive data under disclosure of aggregated data is far from what has been so far believed. Our results therefore call for the re-examination of aggregation and disclosure practices and for further research on the issues exposed herein. Our constructions also provides a powerful automatic tool in constructing concrete examples, such as the possibly smallest 2-margins for (6, 4, 3)-tables with entry-range containing a gap.