### Abstract

We introduce a technique for establishing and amplifying gaps between parameters of network coding and index coding problems. The technique uses linear programs to establish separations between combinatorial and coding-theoretic parameters and applies hyper graph lexicographic products to amplify these separations. This entails combining the dual solutions of the lexicographic multiplicands and proving that this is a valid dual solution of the product. Our result is general enough to apply to a large family of linear programs. This blend of linear programs and lexicographic products gives a recipe for constructing hard instances in which the gap between combinatorial or coding-theoretic parameters is polynomially large. We find polynomial gaps in cases in which the largest previously known gaps were only small constant factors or entirely unknown. Most notably, we show a polynomial separation between linear and non-linear network coding rates. This involves exploiting a connection between matroids and index coding to establish a previously unknown separation between linear and non-linear index coding rates. We also construct index coding problems with a polynomial gap between the broadcast rate and the trivial lower bound for which no gap was previously known.

Original language | English (US) |
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Title of host publication | Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 |

Pages | 609-618 |

Number of pages | 10 |

DOIs | |

State | Published - 2011 |

Event | 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 - Palm Springs, CA, United States Duration: Oct 22 2011 → Oct 25 2011 |

### Other

Other | 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 |
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Country | United States |

City | Palm Springs, CA |

Period | 10/22/11 → 10/25/11 |

### Fingerprint

### Keywords

- index coding
- information inequalities
- lexicographic products
- matroids
- network coding

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011*(pp. 609-618). [6108222] https://doi.org/10.1109/FOCS.2011.39

**Lexicographic products and the power of non-linear network coding.** / Blasiak, Anna; Kleinberg, Robert; Lubetzky, Eyal.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011.*, 6108222, pp. 609-618, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, United States, 10/22/11. https://doi.org/10.1109/FOCS.2011.39

}

TY - GEN

T1 - Lexicographic products and the power of non-linear network coding

AU - Blasiak, Anna

AU - Kleinberg, Robert

AU - Lubetzky, Eyal

PY - 2011

Y1 - 2011

N2 - We introduce a technique for establishing and amplifying gaps between parameters of network coding and index coding problems. The technique uses linear programs to establish separations between combinatorial and coding-theoretic parameters and applies hyper graph lexicographic products to amplify these separations. This entails combining the dual solutions of the lexicographic multiplicands and proving that this is a valid dual solution of the product. Our result is general enough to apply to a large family of linear programs. This blend of linear programs and lexicographic products gives a recipe for constructing hard instances in which the gap between combinatorial or coding-theoretic parameters is polynomially large. We find polynomial gaps in cases in which the largest previously known gaps were only small constant factors or entirely unknown. Most notably, we show a polynomial separation between linear and non-linear network coding rates. This involves exploiting a connection between matroids and index coding to establish a previously unknown separation between linear and non-linear index coding rates. We also construct index coding problems with a polynomial gap between the broadcast rate and the trivial lower bound for which no gap was previously known.

AB - We introduce a technique for establishing and amplifying gaps between parameters of network coding and index coding problems. The technique uses linear programs to establish separations between combinatorial and coding-theoretic parameters and applies hyper graph lexicographic products to amplify these separations. This entails combining the dual solutions of the lexicographic multiplicands and proving that this is a valid dual solution of the product. Our result is general enough to apply to a large family of linear programs. This blend of linear programs and lexicographic products gives a recipe for constructing hard instances in which the gap between combinatorial or coding-theoretic parameters is polynomially large. We find polynomial gaps in cases in which the largest previously known gaps were only small constant factors or entirely unknown. Most notably, we show a polynomial separation between linear and non-linear network coding rates. This involves exploiting a connection between matroids and index coding to establish a previously unknown separation between linear and non-linear index coding rates. We also construct index coding problems with a polynomial gap between the broadcast rate and the trivial lower bound for which no gap was previously known.

KW - index coding

KW - information inequalities

KW - lexicographic products

KW - matroids

KW - network coding

UR - http://www.scopus.com/inward/record.url?scp=84863334401&partnerID=8YFLogxK

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U2 - 10.1109/FOCS.2011.39

DO - 10.1109/FOCS.2011.39

M3 - Conference contribution

AN - SCOPUS:84863334401

SN - 9780769545714

SP - 609

EP - 618

BT - Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011

ER -