[Paper Review] On the diameter of lattice polytopes

Alberto Del Pia, Carla Michini|arXiv (Cornell University)|Dec 24, 2015

Computational Geometry and Mesh Generation11 references4 citations

TL;DR

This paper establishes an improved upper bound on the diameter of $d$-dimensional lattice polytopes contained in $[0,k]^d$, showing it is at most $\left\lfloor\left(k - \frac{1}{2}\right)d\right\rfloor$. The proof uses induction, vertex projection, and structural analysis of faces, and the bound is tight for $k=2$, yielding a diameter of $\left\lfloor\frac{3}{2}d\right\rfloor$ for half-integral polytopes.

ABSTRACT

In this paper we show that the diameter of a d-dimensional lattice polytope in [0,k]^n is at most (k - 1/2) d. This result implies that the diameter of a d-dimensional half-integral polytope is at most 3/2 d. We also show that for half-integral polytopes the latter bound is tight for any d.

Motivation & Objective

• To establish a tighter upper bound on the diameter of $d$-dimensional lattice polytopes contained in $[0,k]^d$.
• To refine the previously known bound of $kd$ by Kleinschmidt and Onn, especially for $k \geq 2$.
• To show that the bound $\left\lfloor\left(k - \frac{1}{2}\right)d\right\rfloor$ is tight for $k=2$ and all $d$.
• To extend the result to half-integral polytopes, which are affinely equivalent to $(0,2)$-polytopes.
• To investigate the asymptotic behavior of the maximum diameter $\delta_k^d$ and identify gaps between known lower and upper bounds.

Proposed method

• Uses induction on dimension $d$, with base cases for $d=1$ and $d=2$.
• Applies vertex projection to reduce a $d$-dimensional polytope to a full-dimensional one in $\mathbb{R}^d$, preserving the 1-skeleton.
• Employs face projections and minimization over linear functionals to bound distances via lower-dimensional polytopes.
• Applies Lemma 3 to bound path lengths across faces using the minimum value of $x_i + x_j$ on a face.
• Uses Lemma 4 to bound the diameter of projected faces in terms of $\delta_k^{d-2}$ and the minimum sum $\gamma$.
• Performs case analysis based on vertex positions relative to the hypercube $[0,k]^d$, using symmetry via variable substitution $\tilde{x}_i = k - x_i$.

Experimental results

Research questions

• RQ1 Can the diameter of a $d$-dimensional $(0,k)$-polytope be bounded more tightly than $kd$?
• RQ2 Is the bound $\left\lfloor\left(k - \frac{1}{2}\right)d\right\rfloor$ tight for $k=2$?
• RQ3 What is the maximum diameter of a half-integral polytope, and how does it relate to $(0,2)$-polytopes?
• RQ4 What is the asymptotic growth of $\delta_k^d$, the maximum diameter over $d$-dimensional $(0,k)$-polytopes?
• RQ5 Can the current upper bound be improved for $k \geq 3$, or is there a fundamental gap in the approach?

Key findings

• The diameter of any $d$-dimensional $(0,k)$-polytope is at most $\left\lfloor\left(k - \frac{1}{2}\right)d\right\rfloor$, improving the prior bound of $kd$.
• For $k=2$, the bound $\left\lfloor\frac{3}{2}d\right\rfloor$ is tight for all $d$, as demonstrated by explicit constructions using cartesian products.
• The polytope $H_d = (H_2)^{d/2}$ for even $d$ and $H_d = H_{d-1} \times H_1$ for odd $d$ achieves diameter $\left\lfloor\frac{3}{2}d\right\rfloor$.
• Half-integral polytopes, whose vertices lie in $\{0, \frac{1}{2}, 1\}^d$, have diameter at most $\left\lfloor\frac{3}{2}d\right\rfloor$, matching the bound for $(0,2)$-polytopes.
• The best known lower bound for $\delta_k^d$ is $\Omega(k^{2/3}d)$, derived from 2D $(0,k)$-polytopes with $\Theta(k^{2/3})$ vertices.
• An asymptotic gap remains: the best upper bound is linear in both $k$ and $d$, while the best known lower bound is $k^{2/3}$-scaled, indicating room for improvement.