A graph theoretic approach to statistical data security
SIAM Journal on Computing
Three-Dimensional Statistical Data Security Problems
SIAM Journal on Computing
Gröbner bases and polyhedral geometry of reducible and cyclic models
Journal of Combinatorial Theory Series A
Random walks on the vertices of transportation polytopes with constant number of sources
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Rapidly Mixing Markov Chains for Sampling Contingency Tables with a Constant Number of Rows
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Bounds on Entries in 3-Dimensional Contingency Tables Subject to Given Marginal Totals
Inference Control in Statistical Databases, From Theory to Practice
Higher Lawrence configurations
Journal of Combinatorial Theory Series A
The Complexity of Three-Way Statistical Tables
SIAM Journal on Computing
Privacy, accuracy, and consistency too: a holistic solution to contingency table release
Proceedings of the twenty-sixth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
The largest group of invariance for Markov bases and toric ideals
Journal of Symbolic Computation
Graphs of transportation polytopes
Journal of Combinatorial Theory Series A
Support sets in exponential families and oriented matroid theory
International Journal of Approximate Reasoning
Entry uniqueness in margined tables
PSD'06 Proceedings of the 2006 CENEX-SDC project international conference on Privacy in Statistical Databases
Discrete Optimization
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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.