### 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 [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 G_{0} 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 G_{1},G_{2} 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.

Original language | English (US) |
---|---|

Title of host publication | 8th Innovations in Theoretical Computer Science Conference, ITCS 2017 |

Editors | Christos H. Papadimitriou |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770293 |

DOIs | |

State | Published - Nov 1 2017 |

Event | 8th Innovations in Theoretical Computer Science Conference, ITCS 2017 - Berkeley, United States Duration: Jan 9 2017 → Jan 11 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 67 |

ISSN (Print) | 1868-8969 |

### Other

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

Country | United States |

City | Berkeley |

Period | 1/9/17 → 1/11/17 |

### All Science Journal Classification (ASJC) codes

- Software

### Keywords

- Gap Amplification
- Multicommodity flows
- Network coding

## Fingerprint Dive into the research topics of 'Coding in undirected graphs is either very helpful or not helpful at all'. Together they form a unique fingerprint.

## Cite this

*8th Innovations in Theoretical Computer Science Conference, ITCS 2017*(Leibniz International Proceedings in Informatics, LIPIcs; Vol. 67). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ITCS.2017.18