Edge removal in undirected networks

TL;DR

This paper proves that removing a $$-capacity edge in undirected networks decreases communication rates proportionally, establishing the equality of zero-error and vanishing-error capacity regions for such networks.

Contribution

It demonstrates that in undirected networks, edge removal reduces capacity by a linear factor, linking zero-error and vanishing-error capacity regions.

Findings

Removing a $$-capacity edge decreases capacity by $O()$

Zero-error and vanishing-error capacity regions are equal in undirected networks

Open question remains for directed networks

Abstract

The edge-removal problem asks whether the removal of a $\lambda$-capacity edge from a given network can decrease the communication rate between source-terminal pairs by more than $\lambda$. In this short manuscript, we prove that for undirected networks, removing a $\lambda$-capacity edge decreases the rate by $O(\lambda)$. Through previously known reductive arguments, here newly applied to undirected networks, our result implies that the zero-error capacity region of an undirected network equals its vanishing-error capacity region. Whether it is possible to prove similar results for directed networks remains an open question.