Reflexive polytopes and discrete polymatroids

TL;DR

This paper classifies discrete polymatroids with independence polytopes that are reflexive, contributing to the understanding of the geometric properties of these combinatorial structures.

Contribution

It provides a classification of discrete polymatroids based on the reflexivity of their independence polytopes, a novel geometric perspective.

Findings

Identifies conditions for reflexivity in independence polytopes

Classifies discrete polymatroids with reflexive polytopes

Enhances understanding of geometric properties of polymatroids

Abstract

A classification of discrete polymatroids whose independence polytopes are reflexive will be presented.

Equations

$$x_i\ge 0,i=1,2,\dots ,d,$$

$$\sum _{i\in A}{x}_i\le |A|+1,A\in \mathcal{A}.$$

$$\bigcup _{A\in \mathcal{A}}A=[d].$$

$$\mathbf{v}=\frac{q}{q-1}\sum _{j=1}^q{\mathbf{e}}_{i_j}\in \mathcal{P}$$

$$\rho (X)\ge |\mathbf{v}|=\frac{q^2}{q-1}>q+1.$$

$$\frac{q^2}{q-1}\le q+2=|X|+2\le |A|+1.$$

$$k\frac{q}{q-1}\le k+1\le |A|+1.$$

$$\rho (A)+\rho (B)=|A|+|B|+2=|A\cup B|+|A\cap B|+2<\rho (A\cup B)+\rho (A\cup B),$$

$$\rho (X)=\min \{|A|+1:X\subseteq A,A\in \mathcal{L}\},\varnothing \ne X\subset [d]$$

$$x_1+x_2=3,x_2+x_3=3,x_1+x_2+x_3=4.$$

$$\mathcal{L}=\{\varnothing ,\{d\},\{d-1,d\},\dots ,\{1,\dots ,d\}\}.$$

$$\mathcal{B}=\{[d],[d],[d-1],\dots ,[2],[1]\}.$$

$${\rho }_P(X)=|\{i,i+1,\dots ,d\}|+1=(d-(i-1))+1=d+2-i$$

$${\rho }_{P^{\prime }}(X)=(d+1)-(i-1)=d+2-i.$$

Full text

The second author was supported by JSPS KAKENHI 19H00637.

Introduction

The discrete polymatroid is introduced in [1]. In the present paper, as a supplement to [1], a classification of discrete polymatroids whose independence polytopes are reflexive will be presented. We refer the reader to [1] and [2] for fundamental materials on discrete polymatroids.

1. Reflexive polytopes

A convex polytope ${\mathcal{P}}\subset{\mathbb{R}}^{d}$ of dimension $d$ is called a lattice polytope if each of its vertices belongs to ${\mathbb{Z}}^{d}$. A reflexive polytope is a lattice polytope ${\mathcal{P}}\subset{\mathbb{R}}^{d}$ of dimension $d$ for which the origin of ${\mathbb{R}}^{d}$ belongs to the interior of ${\mathcal{P}}$ and the dual polytope ${\mathcal{P}}^{\vee}=\{{\mathbf{x}}\in{\mathbb{R}}^{d}:\langle{\mathbf{x}},{\mathbf{y}}\rangle\leq 1,\forall{\mathbf{y}}\in{\mathcal{P}}\}$ of ${\mathcal{P}}$ is a lattice polytope, where $\langle{\mathbf{x}},{\mathbf{y}}\rangle$ stands for the canonical inner product of ${\mathbb{R}}^{d}$. A lattice polytope which can be a reflexive polytope by parallel shift is also called reflexive.

Let ${\mathbf{e}}_{1},\ldots,{\mathbf{e}}_{d}$ denote the canonical basis vectors of ${\mathbb{R}}^{d}$. Let $P\subset{\mathbb{Z}}^{d}_{+}$ be a discrete polymatroid [1, Definition 2.1] on the ground set $[d]$. In what follows one assumes that each ${\mathbf{e}}_{i}$ belongs to $P$. Let ${\mathcal{P}}={\mathcal{P}}_{P}\subset{\mathbb{R}}^{d}$ denote the lattice polytope which is the convex hull of $P$ in ${\mathbb{R}}^{d}$. We call ${\mathcal{P}}$ the independence polytope of $P$. One has $\dim{\mathcal{P}}=d$. Let $\rho=\rho_{P}$ denote the ground set rank function [1, pp. 243] of ${\mathcal{P}}$. It follows from [1, Theorem 7.3] that

Lemma 1.1**.**

The independence polytope ${\mathcal{P}}$ is reflexive if and only if, for each subset $X\subset[d]$ which is $\rho$-closed and $\rho$-inseparable [1, pp. 257–258], one has $\rho(X)=|X|+1$.

A sublattice of $2^{[d]}$ is a collection ${\mathcal{L}}$ of subsets of $[d]$ with $\emptyset\in{\mathcal{L}}$ and $[d]\in{\mathcal{L}}$ such that, for all $A$ and $B$ belonging to ${\mathcal{L}}$, one has $A\cap B\in{\mathcal{L}}$ and $A\cup B\in{\mathcal{L}}$.

Theorem 1.2**.**

(a) Let $P$ be a discrete polymatroid on the ground set $[d]$ and $\rho=\rho_{P}$ the ground set rank function of ${\mathcal{P}}$. Let ${\mathcal{A}}$ be the set of $\rho$-closed and $\rho$-inseparable subsets of ${\mathcal{P}}$. If ${\mathcal{P}}$ is reflexive, then ${\mathcal{A}}\cup\{\emptyset\}$ is a sublattice of $2^{[d]}$.

(b) Conversely, given a sublattice ${\mathcal{L}}$ of $2^{[d]}$, there exists a unique discrete polymatroid $P$ on the ground set $[d]$ for which ${\mathcal{L}}$ is the set of $\rho$-closed and $\rho$-inseparable subsets of ${\mathcal{P}}$ and ${\mathcal{P}}$ is reflexive.

Proof.

(a) If the independence polytope ${\mathcal{P}}$ of $P$ is reflexive, then Lemma 1.1 says that $\rho(A)=|A|+1$ for each $A\in{\mathcal{A}}$. It follows from [1, Proposition 7.2] that ${\mathcal{P}}$ consists of those $(x_{1},\ldots,x_{d})\in{\mathbb{R}}^{d}$ for which

[TABLE]

and

[TABLE]

Since each ${\mathbf{e}}_{i}\in{\mathcal{P}}$ and ${\mathcal{P}}$ is compact, it follows that

[TABLE]

Furthermore, if $X\not\in{\mathcal{A}}$, then $\rho(X)>|X|+1$.

In fact, if $|X|$ = 1 and $X=\{i\}$, then $3{\mathbf{e}}_{i}\in{\mathcal{P}}$ and $\rho(X)>2$. In general, if $|X|=q\geq 2$ and $X=\{i_{1},\ldots,i_{q}\}$ with $i_{1}<\cdots<i_{q}$, then, one has

[TABLE]

and

[TABLE]

To see why (2) holds, one shows that ${\mathbf{v}}$ satisfies each of the inequalities (1). Let $A\in{\mathcal{A}}$ with $X\subsetneq A$, then

[TABLE]

Let $A\in{\mathcal{A}}$ with $|X\cap A|=k<q$, then

[TABLE]

One claims that ${\mathcal{A}}\cup\{\emptyset\}$ is a sublattice of $2^{[d]}$. Let $A,B\in{\mathcal{A}}$ and suppose that either $A\cup B\not\in{\mathcal{A}}\cup\{\emptyset\}$ or $A\cap B\not\in{\mathcal{A}}\cup\{\emptyset\}$. Then

[TABLE]

which contradict the fact that $\rho$ is submodular. Furthermore, since $\bigcup_{A\in{\mathcal{A}}}A=[d]$, one has $[d]\in{\mathcal{A}}$, as desired.

(b) By virtue of [1, Theorem 9.1] one introduces the nondecreasing submodular function $\rho:2^{[n]}\rightarrow{\mathbb{Z}}_{+}$ by setting

[TABLE]

together with $\rho(\emptyset)=0$. Let $P$ be the discrete polymatroid on the ground set $[n]$ and $\rho$ the ground set rank function of ${\mathcal{P}}$. Then ${\mathcal{L}}$ is the set of $\rho$-closed and $\rho$-inseparable subsets of $[d]$. Furthermore, Lemma 1.1 guarantees that the independence polytope ${\mathcal{P}}$ of $P$ is reflexive. On the other hand, suppose that $P^{\prime}$ is a discrete polymatroid on the ground set $[d]$ and $\rho^{\prime}$ its ground set rank function of the independence polytope ${\mathcal{P}}^{\prime}$ of $P^{\prime}$ for which ${\mathcal{L}}$ is the set of $\rho^{\prime}$-closed and $\rho^{\prime}$-inseparable subsets of $P^{\prime}$ and for which ${\mathcal{P}}^{\prime}$ is reflexive. Then by using Lemma 1.1 again one has $\rho^{\prime}(A)=|A|+1$ for each $A\in{\mathcal{L}}$. Hence ${\mathcal{P}}={\mathcal{P}}^{\prime}$ ([1, Proposition 7.2]). Thus $P=P^{\prime}$ ([1, Theorem 3.4]), as desired.

2. Examples

Example 2.1**.**

Let ${\mathcal{P}}\subset{\mathbb{R}}^{3}$ be the convex polytope whose facets are each $x_{i}=0$ together with

[TABLE]

It can be checked that ${\mathcal{P}}$ is reflexive. However, ${\mathcal{P}}$ cannot be the independence polytope of a discrete polymatroid on the ground set $[3]$. In fact, if ${\mathcal{P}}$ is the independence polytope of a discrete polymatroid $P$ on the ground set $[3]$, then both $u=(0,3,0)$ and $v=(1,2,1)$ belong to the set of bases [1, p. 245] of $P$. One has $|u|<|v|$, which contradicts [1, Theorem 2.3]. **

Example 2.2**.**

Let ${\mathcal{L}}$ be a chain of length $d$ of $2^{[d]}$, say,

[TABLE]

Let $P$ denote the discrete polymatroid constructed in Theorem 1.2 (b). Let

[TABLE]

Let $P^{\prime}$ denote the transversal polymatroid [1, p. 267] presented by ${\mathcal{B}}$. If $X\subset[d]$ and $i=\min(X)$, then it follows from the proof of Theorem 1.2 (b) that

[TABLE]

On the other hand, by the definition of the ground set rank function of a transversal polymatroid, one has

[TABLE]

Hence $\rho_{P}=\rho_{P^{\prime}}$. Thus $P=P^{\prime}$. **

It would be of interest for which sublattice ${\mathcal{L}}$ of $2^{[d]}$ the discrete polymatroid constructed in Theorem 1.2 (b) can be a transversal polymatroid.