整数线性不可行问题的推广
A generalization of the integer linear infeasibility problem
摘要：
给定的非负约束线性方程组是否有整数解？这是许多领域的一个基本问题。在统计学中，这个问题出现在列联表数据的数据安全问题中，并且与具有给定边缘的列联表的抽样的马尔可夫基的非平方自由元素密切相关。为了研究一类无整数解的系统，我们研究了由$\Z^d$的有限子集生成的交换半群及其饱和性。在半群的差和它的饱和中的元素称为“洞”。给出了洞集有限的充要条件。定义了交换半群的基本洞和饱和点。然后，我们证明了洞集，非饱和点集和饱和点生成元集的同时有限性。我们将我们的结果应用到一些三向和四向列联表。然后，我们将讨论我们的算法的时间复杂度。
英文摘要：
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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