Computing holes in semi-groups and its applications to transportation problems

An integer feasibility problem is a fundamental problem in many areas, such as operations research, number theory, and statistics. To study a family of systems with no nonnegative integer solution, we focus on a commutative semigroup generated by a finite set of vectors in Z and its saturation. In this paper we present an algorithm to compute an explicit description for the set of holes which is the difference of a semi-group Q generated by the vectors and its saturation. We apply our procedure to compute an infinite family of holes for the semi-group of the 3× 4× 6 transportation problem. Furthermore, we give an upper bound for the entries of the holes when the set of holes is finite. Finally, we present an algorithm to find all Q-minimal saturation points of Q.