Project Euclid

The question whether there exists an integral solution to the system of linear equations with non-negativity constraints, $A\mathbf{x} = \mathbf{b}, \, \mathbf{x} \ge 0$, where $A \in \mathbb{Z}^{m\times n}$ and ${\mathbf b} \in \mathbb{Z}^m$, finds its applications in many areas such as operations research, number theory, combinatorics, and statistics. In order to solve this problem, we have to understand the semigroup generated by the columns of the matrix $A$ and the structure of the “holes” which are the difference between the semigroup and its saturation. In this paper, we discuss the implementation of an algorithm by Hemmecke, Takemura, and Yoshida that computes the set of holes of a semigroup and we discuss applications to problems in combinatorics. Moreover, we compute the set of holes for the common diagonal effect model and we show that the $n^\text{th}$ linear ordering polytope has the integer-decomposition property for $n\leq 7$. The software is available at http://ehrhart.math.fu-berlin.de/People/fkohl/HASE/.