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Revision #1 to TR16-110 | 16th February 2017 14:36

The Minrank of Random Graphs

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Revision #1
Authors: Alexander Golovnev, Oded Regev, Omri Weinstein
Accepted on: 16th February 2017 14:36
Downloads: 103
Keywords: 


Abstract:

The minrank of a graph $G$ is the minimum rank of a matrix $M$ that can be obtained from the adjacency matrix of $G$ by
switching some ones to zeros (i.e., deleting edges) and then setting all diagonal entries to one.
This quantity is closely related to the fundamental information-theoretic problems of (linear) index coding
(Bar-Yossef et al., FOCS'06),
network coding and distributed storage, and to Valiant's approach for proving superlinear circuit lower bounds (Valiant, Boolean Function Complexity '92).

We prove tight bounds on the minrank of random Erd\H{o}s-R\'enyi graphs $G(n,p)$ for all regimes of $p\in[0,1]$. In particular, for any constant $p$, we show that $\mathsf{minrk}(G) = \Theta(n/\log n)$
with high probability, where $G$ is chosen from $G(n,p)$.
This bound gives a near quadratic improvement over the previous best lower bound of $\Omega(\sqrt{n})$ (Haviv and Langberg, ISIT'12),
and partially settles an open problem raised by Lubetzky and Stav (FOCS '07).
Our lower bound matches the well-known upper bound obtained by the "clique covering"
solution, and settles the linear index coding problem for random graphs.

Finally, our result suggests a new avenue of attack, via derandomization, on
Valiant's approach for proving superlinear lower bounds for logarithmic-depth semilinear circuits.


Paper:

TR16-110 | 19th July 2016 10:02

The Minrank of Random Graphs





TR16-110
Authors: Alexander Golovnev, Oded Regev, Omri Weinstein
Publication: 20th July 2016 07:29
Downloads: 332
Keywords: 


Abstract:

The minrank of a graph $G$ is the minimum rank of a matrix $M$ that can be obtained from the adjacency matrix of $G$ by switching ones to zeros (i.e., deleting edges) and setting all diagonal entries to one. This quantity is closely related to the fundamental information-theoretic problems of (linear) index coding
(Bar-Yossef et al., FOCS'06), network coding and distributed storage, and to Valiant's approach for proving superlinear circuit lower bounds (Valiant, Boolean Function Complexity '92).

We prove tight bounds on the minrank of random Erd\H{o}s-R\'enyi graphs $G(n,p)$ for all regimes of $p\in[0,1]$. In particular, for any constant $p$, we show that $minrk(G) = \Theta(n/\log n)$ with high probability, where $G$ is chosen from $G(n,p)$.

This bound gives a near quadratic improvement over the previous best lower bound of $\Omega(\sqrt{n})$ (Haviv and Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky and Stav (FOCS '07). Our lower bound matches the well-known upper bound obtained by the "clique covering" solution, and settles the linear index coding problem for random knowledge graphs.

Finally, our result suggests a new avenue of attack, via derandomization, on Valiant's approach for proving superlinear lower bounds for logarithmic-depth semilinear circuits.



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