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

An arbitrary starting homotopy-like simplicial algorithm is developed for computing an integer point in a polytope given by $P=\{x\mid Ax\leq b\}$ satisfying that each row of $A$ has at most one positive entry. The algorithm is derived from an introduction of an integer labeling rule and an application of a triangulation of the space $R^n\times[0,1]$. It consists of two phases, one of which forms an $(n+1)$-dimensional pivoting procedure and the other an $n$-dimensional pivoting procedure. Starting from an arbitrary integer point in $R^n\times\{0\}$, the algorithm interchanges from one phase to the other, if necessary, and follows a finite simplicial path that either leads to an integer point in the polytope or proves that no such point exists.