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

Abstract

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 $\Z^d$ 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 $3\times 4\times 6$ 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 $Q$.