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) |
|---|---|
| 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 |
|---|---|
| Country | United States |
| City | Palm Springs, CA |
| Period | 10/22/11 → 10/25/11 |
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Keywords
- index coding
- information inequalities
- lexicographic products
- matroids
- network coding
ASJC Scopus subject areas
- Computer Science(all)
Cite this
Lexicographic products and the power of non-linear network coding. / Blasiak, Anna; Kleinberg, Robert; Lubetzky, Eyal.
Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011. 2011. p. 609-618 6108222.Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
}
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
UR - http://www.scopus.com/inward/citedby.url?scp=84863334401&partnerID=8YFLogxK
U2 - 10.1109/FOCS.2011.39
DO - 10.1109/FOCS.2011.39
M3 - Conference contribution
SN - 9780769545714
SP - 609
EP - 618
BT - Proceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011
ER -