TY - GEN

T1 - Coding in undirected graphs is either very helpful or not helpful at all

AU - Braverman, Mark

AU - Garg, Sumegha

AU - Schvartzman, Ariel

PY - 2017/11/1

Y1 - 2017/11/1

N2 - 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 [11] 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 G0 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.

AB - 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 [11] 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 G0 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.

KW - Gap Amplification

KW - Multicommodity flows

KW - Network coding

UR - http://www.scopus.com/inward/record.url?scp=85038593618&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85038593618&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ITCS.2017.18

DO - 10.4230/LIPIcs.ITCS.2017.18

M3 - Conference contribution

AN - SCOPUS:85038593618

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 8th Innovations in Theoretical Computer Science Conference, ITCS 2017

A2 - Papadimitriou, Christos H.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 8th Innovations in Theoretical Computer Science Conference, ITCS 2017

Y2 - 9 January 2017 through 11 January 2017

ER -