A generalization of the integer linear infeasibility problem

Akimichi Takemura; Ruriko Yoshida

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A generalization of the integer linear infeasibility problem

Akimichi Takemura·Ruriko Yoshida

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Abstract

Does a given system of linear equations with nonnegative constraints have an integer solution? This is a fundamental question in many areas. In statistics this problem arises in data security problems for contingency table data and also is closely related to non-squarefree elements of Markov bases for sampling contingency tables with given marginals. To study a family of systems with no integer solution, we focus on a commutative semigroup generated by a finite subset of $\Z^d$ and its saturation. An element in the difference of the semigroup and its saturation is called a ``hole''. We show the necessary and sufficient conditions for the finiteness of the set of holes. Also we define fundamental holes and saturation points of a commutative semigroup. Then, we show the simultaneous finiteness of the set of holes, the set of non-saturation points, and the set of generators for saturation points. We apply our results to some three- and four-way contingency tables. Then we will discuss the time complexities of our algorithms.

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Akimichi Takemura, Ruriko Yoshida. 2008-04-12. A generalization of the integer linear infeasibility problem. https://arxiv.org/abs/math/0603108

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