### 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 information-theoretic problems of (linear) index coding (Bar-Yossef et al., IEEE Trans. Inf. Theory 2011), network coding (Effros et al., IEEE Trans. Inf. Theory 2015) and distributed storage (Mazumdar, ISIT, 2014). We prove tight bounds on the minrank of directed Erdős-Rényi random graphs G(n,p) for all regimes of p ∈ [0, 1]. In particular, for any constant p, we show that minrk(G) = Θ(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 Ω( √n) (Haviv and Langberg, ISIT 2012), and partially settles an open problem raised by Lubetzky and Stav (IEEE Trans. Inf. Theory 2009). 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.

Original language | English (US) |
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Journal | IEEE Transactions on Information Theory |

DOIs | |

State | Accepted/In press - Feb 27 2018 |

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### Keywords

- Channel coding
- Complexity theory
- Index coding
- Indexes
- Linear index coding
- Minrank
- Network coding
- Receivers
- Sparse matrices

### ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Library and Information Sciences

### Cite this

*IEEE Transactions on Information Theory*. https://doi.org/10.1109/TIT.2018.2810384