A New Arbitrary Starting Simplicial Algorithm for Computing an Integer Point of an n-Dimensional Simplex∗

Determining whether there is an integer point in an n-dimensional simplex is an NPcomplete problem. In this paper, a new arbitrary starting variable dimension algorithm is developed for computing an integer point of an n-dimensional simplex. The algorithm is derived from an introduction of an integer labeling rule and an application of a triangulation of the space and is composed of two phases, one of which forms a variable dimension algorithm and the other a fulldimension pivoting procedure. Starting at an arbitrary integer point, the algorithm interchanges from one phase to the other if necessary and follows a finite simplicial path that either leads to an integer point of the simplex or proves that no such points exist. An advantage of the algorithm is that all the existing superior triangulations can be its underlying triangulations without any modification.