Recognising Graphic and Matroidal Connectivity Functions

TL;DR

This paper presents a polynomial-time method to recognize when a connectivity function originates from a graph, but proves that identifying matroidal connectivity functions is computationally hard.

Contribution

It introduces a polynomial-time approach for recognizing graph connectivity functions and establishes the computational hardness of recognizing matroidal connectivity functions.

Findings

Polynomial-time method for recognizing graph connectivity functions

Recognition of matroidal connectivity functions is NP-hard

Recognition of non-matroidal connectivity functions is NP-hard

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. In this paper we give a method for identifying when a connectivity function comes from a graph. This method uses no more than a polynomial number of evaluations of the connectivity function. In contrast, we show that the problem of identifying when a connectivity function comes from a matroid cannot be solved in polynomial time. We also show that the problem of identifying when a connectivity function is not that of a matroid cannot be solved in polynomial time.