A generalization of the integer linear infeasibility problem

Does a given system of linear equations Ax = b have a nonnegative integer solution? This is a fundamental question in many areas, such as operations research, number theory, and statistics. In terms of optimization, this is called an integer feasibility problem. A generalized integer feasibility problem is to find b such that there does not exist a nonnegative integral solution in the system with a given A. One such problem is the well-known Frobenius problem. In this paper we study the generalized integer feasibility problem and also the multi-dimensional Frobenius problem. To study a family of systems with no nonnegative integer solution, we focus on a commutative semigroup generated by a finite subset of Z 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. As examples we consider some threeand fourway contingency tables from statistics and apply our results to them. Then we will discuss the time complexities of our algorithms.