- Published 2016 in ArXiv

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 in [9] 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 + ε) 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 G1, G2 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. ∗mbraverm@princeton.edu. Supported in part by an NSF CAREER award (CCF-1149888), NSF CCF-1525342, a Packard Fellowship in Science and Engineering, and the Simons Collaboration on Algorithms and Geometry. †sumeghag@cs.princeton.edu ‡acohenca@cs.princeton.edu 1 ar X iv :1 60 8. 06 54 5v 1 [ cs .I T ] 2 3 A ug 2 01 6

@article{Braverman2016NetworkCI,
title={Network coding in undirected graphs is either very helpful or not helpful at all},
author={Mark Braverman and Sumegha Garg and Ariel Schvartzman},
journal={CoRR},
year={2016},
volume={abs/1608.06545}
}