随机图的最小秩
The Minrank of Random Graphs
摘要：
图$G$的minrank是矩阵$M$的最小秩，其可以通过将一些1切换为0（即，删除边），然后将所有对角线条目设置为1。这个量与（线性）索引编码的基本信息理论问题密切相关（Bar-Yossef等人，FOCS'06）、网络编码和分布式存储，以及Valiant的用于证明超线性电路下界的方法（Valiant，Boolean Function Complexity '92）。我们证明了紧界上的最小秩的随机Erdens-Rényi图$G(n,p)$的所有政权的$p\in[0,1]$。特别是，对于任何常数$p$，我们表明，$\mathsf{minrk}(G) = Θ(n/\log n)$具有高概率，其中$G$是从$G(n,p)$选择。这个界限给出了一个接近二次的改善，以前的最佳下限的$Ω(\sqrt{n})$（Haviv和Langberg，ISIT'12），并部分解决了一个开放的问题，提出了Lubetzky和Stav（FOCS '07）。我们的下界匹配的“团覆盖”的解决方案，并解决了随机图的线性指数编码问题所获得的著名的上界。最后，我们的结果提出了一种新的攻击途径，通过去随机化，Valiant的方法证明超线性下界的深度半线性电路。
英文摘要：
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 $p\in[0,1]$. In particular, for any constant $p$, we show that $\mathsf{minrk}(G) = Θ(n/\log n)$ 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 $Ω(\sqrt{n})$ (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.
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