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 -