An Arbitrary Starting Homotopy-Like Simplicial Algorithm for Computing an Integer Point in a Class of Polytopes

Let P be a polytope satisfying that each row of the defining matrix has at most one positive entry. Determining whether there is an integer point in P is known to be an NP-complete problem. By introducing an integer labeling rule on an augmented set and applying a triangulation of the Euclidean space, we develop in this paper a variable dimension method for computing an integer point in P. The method starts from an arbitrary integer point and follows a finite simplicial path that either leads to an integer point in P or proves no such point exists.