Small Shadows of Lattice Polytopes

The diameter of the graph of a $d$-dimensional lattice polytope $P \subseteq [0,k]^{n}$ is known to be at most $dk$ due to work by Kleinschmidt and Onn. However, it is an open question whether the monotone diameter, the shortest guaranteed length of a monotone path, of a $d$-dimensional lattice polytope $P = \{\mathbf{x}: A\mathbf{x} \leq \mathbf{b}\} \subseteq [0,k]^{n}$ is bounded by a polynomial in $d$ and $k$. This question is of particular interest in linear optimization, since paths traced by the Simplex method must be monotone. We introduce partial results in this direction including a monotone diameter bound of $3d$ for $k = 2$, a monotone diameter bound of $(d-1)m+1$ for $d$-dimensional $(m+1)$-level polytopes, a pivot rule such that the Simplex method is guaranteed to take at most $dnk||A||_{\infty}$ non-degenerate steps to solve a LP on $P$, and a bound of $dk$ for lengths of paths from certain fixed starting points. Finally, we present a constructive approach to a diameter bound of $(3/2)dk$ and describe how to translate this final bound into an algorithm that solves a linear program by tracing such a path.