The Minrank of Random Graphs

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 Erdős-Rényi 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'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.

The (Minimum) Rank of Typical Fooling-Set Matrices

“…The bound in Thereom 2(b) is similar to that in [8], but it is better by roughly a factor of log n if p is (constant or) slowly decreasing, e.g., p = 1/ log n. (Their minrank definition gives a lower bound to fooling-set pattern minimum rank. )…”

The minrank of random graphs over arbitrary fields

“…Using Azuma's inequality for the vertex exposure martingale, one can show that min-rank F (G) is highly concentrated around its expectation, which is Ω( n log(1/p) log n ) by Theorem 1.2. This way, one can deduce that min-rank F (G) ≥ Ω( n log(1/p) log n ) holds with probability at least 1 − e −Ω(n/ log 2 n) , see [6] for a detailed argument.…”

“…Finally, we will need the following simple lemma from [6] (which follows, with a slightly better constant, from Turán's Theorem).…”

“…• Haviv has recently combined the key lemma of [6] with the Lovász Local Lemma and proved a related result. To state it, we use the following density parameter.…”

“…The analogous problem over finite fields was raised by Lubetzky and Stav [15] also in the context of linear index coding. Haviv and Langberg [10] proved a lower bound of Ω( √ n) for the minrank of G(n, p) over any fixed finite field and for any constant p. In a recent beautiful paper, Golovnev, Regev, and Weinstein [6] substantially improved the aformentioned results by showing that for any finite field F = F(n) and every p = p(n) in (0, 1), the minrank of the random graph G(n, p) over F is with high probability at least Ω n log(1/p) log(n|F|/p) .…”