Network coding in undirected graphs is either very helpful or not helpful at all

TL;DR

This paper investigates the potential advantage of network coding over multicommodity flow in undirected networks, showing that even small advantages can be amplified to near-maximum gaps, thus deepening understanding of network coding's effectiveness.

Contribution

It introduces a gap amplification method for undirected networks, demonstrating how small coding advantages can lead to large gaps close to theoretical upper bounds.

Findings

Small coding advantages can be amplified to near-maximum gaps.

The construction uses a graph tensor product to multiply gaps.

The approach approaches the known upper bound for gaps.

Abstract

While it is known that using network coding can significantly improve the throughput of directed networks, it is a notorious open problem whether coding yields any advantage over the multicommodity flow (MCF) rate in undirected networks. It was conjectured by Li and Li (2004) that the answer is "no". In this paper we show that even a small advantage over MCF can be amplified to yield a near-maximum possible gap. We prove that any undirected network with $k$ source-sink pairs that exhibits a $(1+\varepsilon)$ gap between its MCF rate and its network coding rate can be used to construct a family of graphs $G'$ whose gap is $\log(|G'|)^c$ for some constant $c < 1$. The resulting gap is close to the best currently known upper bound, $\log(|G'|)$, which follows from the connection between MCF and sparsest cuts. Our construction relies on a gap-amplifying graph tensor product that, given two graphs $G_1,G_2$ with small gaps, creates another graph $G$ with a gap that is equal to the product of the previous two, at the cost of increasing the size of the graph. We iterate this process to obtain a gap of $\log(|G'|)^c$ from any initial gap.