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

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 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

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Cite this paper

@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} }