Small Shadows of Lattice Polytopes

TL;DR

This paper investigates the monotone diameter of lattice polytopes, providing new bounds and algorithms related to the paths taken by the Simplex method in linear programming.

Contribution

It introduces novel bounds on the monotone diameter of lattice polytopes and develops an algorithm for solving LPs by tracing these paths.

Findings

Monotone diameter bound of 3d for k=2

Bound of (d-1)m+1 for (m+1)-level polytopes

Simplex method step bound of dn k ||A||_∞

Abstract

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.