Lexicographic Products and the Power of Non-linear Network Coding


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.

DOI: 10.1109/FOCS.2011.39

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@article{Blasiak2011LexicographicPA, title={Lexicographic Products and the Power of Non-linear Network Coding}, author={Anna Blasiak and Robert D. Kleinberg and Eyal Lubetzky}, journal={2011 IEEE 52nd Annual Symposium on Foundations of Computer Science}, year={2011}, pages={609-618} }