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Coding in Undirected Graphs Is Either Very Helpful or Not Helpful at All

Author(s): Braverman, Mark; Garg, Sumegha; Schvartzman, Ariel

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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 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+epsilon) 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.
Publication Date: 2017
Citation: Braverman, Mark, Sumegha Garg, and Ariel Schvartzman. "Coding in Undirected Graphs Is Either Very Helpful or Not Helpful at All." 8th Innovations in Theoretical Computer Science Conference (ITCS) 67 (2017). doi:10.4230/LIPIcs.ITCS.2017.18
DOI: 10.4230/LIPIcs.ITCS.2017.18
ISSN: 1868-8969
Pages: 18:1 - 18:18
Type of Material: Conference Article
Journal/Proceeding Title: 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)
Version: Final published version. This is an open access article.

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