The minrank of random graphs

Alexander Golovnev, Oded Regev, Omri Weinstein

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

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 informationtheoretic 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-Rényi random graphs G(n, p) for all regimes of p 2 [0, 1]. In particular, for any constant p, we show that minrk(G) = (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 (p n) (Haviv and Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky and Stav (FOCS '07). Our lower bound matches the wellknown 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.

Original languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Volume81
ISBN (Electronic)9783959770446
DOIs
StatePublished - Aug 1 2017
Event20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017 - Berkeley, United States
Duration: Aug 16 2017Aug 18 2017

Other

Other20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017
CountryUnited States
CityBerkeley
Period8/16/178/18/17

Fingerprint

Network coding
Boolean functions
Networks (circuits)
Directed graphs

Keywords

  • Circuit complexity
  • Index coding
  • Information theory

ASJC Scopus subject areas

  • Software

Cite this

Golovnev, A., Regev, O., & Weinstein, O. (2017). The minrank of random graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017 (Vol. 81). [46] Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2017.46

The minrank of random graphs. / Golovnev, Alexander; Regev, Oded; Weinstein, Omri.

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017. Vol. 81 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2017. 46.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Golovnev, A, Regev, O & Weinstein, O 2017, The minrank of random graphs. in Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017. vol. 81, 46, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017, Berkeley, United States, 8/16/17. https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2017.46
Golovnev A, Regev O, Weinstein O. The minrank of random graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017. Vol. 81. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2017. 46 https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2017.46
Golovnev, Alexander ; Regev, Oded ; Weinstein, Omri. / The minrank of random graphs. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017. Vol. 81 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2017.
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