Before revision: We consider the problem of optimizing a linear function over a lattice polytope P contained in [0,k]^n and defined via m linear inequalities.
After revision: The goal of this paper is to design a simplex algorithm for linear programs on lattice polytopes that traces `short' simplex paths from any given vertex to an optimal one. We consider a lattice polytope P contained in [0,k]^n and defined via m linear inequalities.