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oai:drops-oai.dagstuhl.de:7595 2024-03-06T10:40:42Z ddc:004 open_access

The Minrank of Random Graphs Golovnev, Alexander Regev, Oded Weinstein, Omri circuit complexity index coding information theory The minrank of a directed 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 directed Erdos-Renyi random graphs G(n,p) for all regimes of 0<p<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 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Alexander Golovnev and Oded Regev and Omri Weinstein 2017 Is Part Of LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX-RANDOM.2017.46 urn:nbn:de:0030-drops-75953 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.46 eng https://creativecommons.org/licenses/by/3.0/legalcode