Connectivity Functions and Polymatroids

TL;DR

This paper explores the properties of connectivity functions associated with polymatroids, introduces a duality concept, and demonstrates that all such functions can be represented by self-dual polymatroids, including integral cases.

Contribution

It introduces a duality notion for polymatroids and proves that all connectivity functions correspond to self-dual polymatroids, extending to integral functions with half-integral self-dual polymatroids.

Findings

Every connectivity function is realizable by a self-dual polymatroid.

Integral connectivity functions correspond to half-integral self-dual polymatroids.

The paper establishes a duality framework for polymatroids.

Abstract

A {\em connectivity function on} a set $E$ is a function $\lambda:2^E\rightarrow \mathbb R$ such that $\lambda(\emptyset)=0$, that $\lambda(X)=\lambda(E-X)$ for all $X\subseteq E$ and that $\lambda(X\cap Y)+\lambda(X\cup Y)\leq \lambda(X)+\lambda(Y)$ for all $X,Y \subseteq E$. Graphs, matroids and, more generally, polymatroids have associated connectivity functions. We introduce a notion of duality for polymatroids and prove that every connectivity function is the connectivity function of a self-dual polymatroid. We also prove that every integral connectivity function is the connectivity function of a half-integral self-dual polymatroid.