Computational determination of the largest lattice polytope diameter

TL;DR

This paper develops a computational approach to determine the maximum diameter of lattice polytopes in small dimensions, confirming conjectures for specific cases in three dimensions.

Contribution

It introduces a computational framework to find the largest lattice polytope diameter and verifies conjectured bounds for (3,4) and (3,5) cases.

Findings

(3,4) = 7

(3,5) = 9

Confirmed the conjecture relating diameter to (k+1)d/2

Abstract

A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let {\delta}(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational framework to determine {\delta}(d, k) for small instances. We show that {\delta}(3, 4) = 7 and {\delta}(3, 5) = 9; that is, we verify for (d, k) = (3, 4) and (3, 5) the conjecture whereby {\delta}(d, k) is at most (k + 1)d/2 and is achieved, up to translation, by a Minkowski sum of lattice vectors.

Tables

Table 1. Table 1: The largest possible diameter δ ​ ( d , k ) 𝛿 𝑑 𝑘 \delta(d,k) of a lattice ( d , k ) 𝑑 𝑘 (d,k) -polytope

	1	2	3	4	5	6	7	8	9	10
1	1	1	1	1	1	1	1	1	1	1
2	2	3	4	4	5	6	6	7	8	8
3	3	4	6	7	9
4	4	6	8
$⋮$	$⋮$	$⋮$
$d$	$d$	$\lfloor \frac{3d}{2}\rfloor$

Full text

††volume: NN††journal: Electronic Notes in Discrete Mathematics

††thanks: Email: \normalshapechaddens@mcmaster.ca††thanks: Email: \normalshapedeza@mcmaster.ca

1 Introduction

Finding a good bound on the maximal edge-diameter of a polytope in terms of its dimension and the number of its facets is not only a natural question of discrete geometry, but also historically closely connected with the theory of the simplex method, as the diameter is a lower bound for the number of pivots required in the worst case. Considering bounded polytopes whose vertices are rational-valued, we investigate a similar question where the number of facets is replaced by the grid embedding size.

The convex hull of integer-valued points is called a lattice polytope and, if all the vertices are drawn from $\{0,1,\dots,k\}^{d}$, it is referred to as a lattice $(d,k)$-polytope. Let $\delta(d,k)$ be the largest edge-diameter over all lattice $(d,k)$-polytopes. Naddef [7] showed in 1989 that $\delta(d,1)=d$, Kleinschmidt and Onn [6] generalized this result in 1992 showing that $\delta(d,k)\leq kd$. In 2016, Del Pia and Michini [3] strengthened the upper bound to $\delta(d,k)\leq kd-\lceil d/2\rceil$ for $k\geq 2$, and showed that $\delta(d,2)=\lfloor 3d/2\rfloor$. Pursuing Del Pia and Michini’s approach, Deza and Pournin [5] showed that $\delta(d,k)\leq kd-\lceil 2d/3\rceil-(k-3)$ for $k\geq 3$, and that $\delta(4,3)=8$. The determination of $\delta(2,k)$ was investigated independently in the early nineties by Thiele [8], Balog and Bárány [2], and Acketa and Žunić [1]. Deza, Manoussakis, and Onn [4] showed that $\delta(d,k)\geq\lfloor(k+1)d/2\rfloor$ for all $k\leq 2d-1$ and proposed Conjecture 1.1.

Conjecture 1.1**.**

$\delta(d,k)\leq\lfloor(k+1)d/2\rfloor$, and $\delta(d,k)$ is achieved, up to translation, by a Minkowski sum of lattice vectors.

In Section 2, we propose a computational framework which drastically reduces the search space for lattice $(d,k)$-polytopes achieving a large diameter. Applying this framework to $(d,k)=(3,4)$ and $(3,5)$, we determine in Section 3 that $\delta(3,4)=7$ and $\delta(3,5)=9$.

Theorem 1.2**.**

Conjecture 1.1 holds for $(d,k)=(3,4)$ and $(3,5)$; that is, $\delta(3,4)=7$ and $\delta(3,5)=9$, and both diameters are achieved, up to translation, by a Minkowski sum of lattice vectors

Note that Conjecture 1.1 holds for all known values of $\delta(d,k)$ given in Table 1, and hypothesizes, in particular, that $\delta(d,3)=2d$. The new entries corresponding to $(d,k)=(3,4)$ and $(3,5)$ are entered in bold.

2 Theoretical and Computational Framework

Since $\delta(2,k)$ and $\delta(d,2)$ are known, we consider in the remainder of the paper that $d\geq 3$ and $k\geq 3$. While the number of lattice $(d,k)$-lattice polytopes is finite, a brute force search is typically intractable, even for small instances. Theorem 2.1, which recalls conditions established in [5], allows to drastically reduce the search space.

Theorem 2.1**.**

For $d\geq 3$, let $d(u,v)$ denote the distance between two vertices $u$ and $v$ in the edge-graph of a lattice $(d,k)$-polytope $P$ such that $d(u,v)=\delta(d,k)$. For $i=1,\dots,d$, let $F_{i}^{0}$, respectively $F_{i}^{k}$, denote the intersection of $P$ with the facet of the cube $[0,k]^{d}$ corresponding to $x_{i}=0$, respectively $x_{i}=k$. Then, $d(u,v)\leq\delta(d-1,k)+k$, and the following conditions are necessary for the inequality to hold with equality:

• $(1)$

$u+v=(k,k,\dots,k)$,

• $(2)$

any edge of $P$ with $u$ or $v$ as vertex is $\{-1,0,1\}$-valued,

• $(3)$

for $i=1,\dots,d$, $F_{i}^{0}$, respectively $F_{i}^{k}$, is a $(d-1)$-dimensional face of $P$ with diameter $\delta(F_{i}^{0})=\delta(d-1,k)$, respectively $\delta(F_{i}^{k})=\delta(d-1,k)$.

Thus, to show that $\delta(d,k)<\delta(d-1,k)+k$, it is enough to show that there is no lattice $(d,k)$-polytope admitting a pair of vertices $(u,v)$ such that $d(u,v)=\delta(d,k)$ and the conditions $(1)$, $(2)$, and $(3)$ are satisfied. The computational framework to determine, given $(d,k)$, whether $\delta(d,k)=\delta(d-1,k)+k$ is outlined below and illustrated for $(d,k)=(3,4)$ or $(3,5)$.

**Algorithm to determine whether $\delta(d,k)<\delta(d-1,k)+k$

***Step $1$: Initialization

Determine the set ${\mathcal{F}}$ of all the lattice $(d-1,k)$-polytopes $P$ such that $\delta(P)=\delta(d-1,k)$. For example, for $(d,k)=(3,4)$, the determination of all the $335$ lattice $(2,4)$-polygons $P$ such that $\delta(P)=4$ is straightforward.

Step $2$: Symmetries

Consider, up to the symmetries of the cube $[0,k]^{d}$, the possible entries for a pair of vertices $(u,v)$ such that $u+v=\{k,k,\dots,k\}$. For example, for $(d,k)=(3,4)$, the following 6 vertices cover all possibilities for $u$ up to symmetry: $(0,0,0),(0,0,1),(0,0,2),(0,1,1),(0,1,2)$, and $(0,2,2)$, where $v=(4,4,4)-u$.

Step $3$: Shelling

For each of the possible pairs $(u,v)$ determined during Step $2$, consider all possible ways for $2d$ elements of the set ${\mathcal{F}}$ determined during Step 1 to form the $2d$ facets of $P$ lying on a facet of the cube $[0,k]^{d}$. For example, for $(d,k)=(3,4)$ and $u=(0,0,0)$, we must find $6$ elements of ${\mathcal{F}}$, $3$ with $(0,0)$ as a vertex, and $3$ with $(4,4)$ as a vertex. In addition, if an edge of an element of ${\mathcal{F}}$ with $u$ or $v$ as vertex is not $\{-1,0,1\}$-valued, this element is disregarded.

 Note that since the choice of an element of ${\mathcal{F}}$ defines the vertices of $P$ belonging to a facet of the cube $[0,k]^{d}$, the choice for the next element of ${\mathcal{F}}$ to form a shelling is significantly restricted. In addition, if the set of vertices and edges belonging to the current elements of ${\mathcal{F}}$ considered for a shelling includes a path from $u$ to $v$ of length at most $\delta(d-1,k)+k-1$, a shortcut between $u$ and $v$ exists and the last added elements of ${\mathcal{F}}$ can be disregarded.

Step $4$. Inner points

For each choice of $2d$ elements of ${\mathcal{F}}$ forming a shelling obtained during Step $3$, consider the $\{1,2,\dots,k-1\}$-valued points not in the convex hull of the vertices of the $2d$ elements of ${\mathcal{F}}$ forming a shelling. Each such $\{1,2,\dots,k-1\}$-valued point is considered as a potential vertex of $P$ in a binary tree. If the current set of edges includes a path from $u$ to $v$ of length at most $\delta(d-1,k)+k-1$, a shortcut between $u$ and $v$ exists and the corresponding node of the binary tree can be disregarded, and the the binary tree is pruned at this node.

A convex hull and diameter computation are performed for each node of the obtained binary tree. If there is a node yielding a diameter of $\delta(d-1,k)+k$ we can conclude that $\delta(d,k)=\delta(d-1,k)+k$. Otherwise, we can conclude that $\delta(d,k)<\delta(d-1,k)+k$. For example, for $(d,k)=(3,5)$, no choice of $6$ elements of ${\mathcal{F}}$ forming a shelling such that $d(u,v)\geq 10$ exist, and thus Step 4 is not executed. *

3 Computational Results

For $(d,k)=(3,4)$, a shelling exists for which path lengths are not decidable by the algorithm without convex hull computations. However, this shelling only achieves a diameter of 7. For $(d,k)=(3,5)$ the algorithm stops at Step $3$, as there is no combination of $6$ elements of ${\mathcal{F}}$ which form a shelling such that $d(u,v)\geq\delta(2,5)+5$. Thus, no convex hull computations are required for $(d,k)=(3,5)$. A shortcut from $u$ to $v$ is typically found early on in the shelling, which leads to the algorithm terminating quickly. Run on a 2009 Intel® Core™2 Duo 2.20GHz CPU, the algorithm is able to terminate for $(d,k)=(3,4)$ and $(3,5)$ in under a minute. Consequently, $\delta(3,4)<8$ and $\delta(3,5)<10$. Since the Minkowski sum of $(1,0,0),(0,1,0),(0,0,1),(0,1,1),(1,0,1),(1,1,0)$, and $(1,1,1)$ forms a lattice $(3,4)$-polytope with diameter $7$, we conclude that $\delta(3,4)=7$. Similarly, since the Minkowski sum of $(1,0,0),(0,1,0),(0,0,1),$ $(0,1,1),(1,0,1),(1,1,0),(0,1,-1),(1,0,-1)$, and $(1,-1,0)$ forms, up to translation, a lattice $(3,5)$-polytope with diameter $9$, we conclude that $\delta(3,5)=9$. Computations for additional values of $\delta(d,k)$ are currently underway. In particular, the same algorithm may determine whether $\delta(d,k)=\delta(d-1,k)+k$ or $\delta(d-1,k)+k-1$ for $(d,k)=(5,3)$ and $(4,4)$ provided the set of all lattice $(d-1,k)$-polytopes achieving $\delta(d-1,k)$ is determined for $(d,k)=(5,3)$ and $(4,4)$. Similarly, the algorithm could be adapted to determine whether $\delta(d,k)<\delta(d-1,k)+k-1$ provided the set of all lattice $(d-1,k)$-polytopes achieving $\delta(d-1,k)$ or $\delta(d-1,k)-1$ is determined. For example, the adapted algorithm may determine whether $\delta(3,6)=10$.

{ack}

This work was partially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant program (RGPIN-2015-06163).