通信复杂性中对数秩猜想的可加组合学方法

Worst-Case to Average-Case Reductions via Additive Combinatorics

We present a new framework for designing worst-case to average-case reductions. For a large class of problems, it provides an explicit transformation of algorithms running in time $T$ that are only correct on a small (subconstant) fraction of their inputs into algorithms running in time $\widetilde{O}(T)$ that are correct on all inputs. Using our framework, we obtain such efficient worst-case to average-case reductions for fundamental problems in a variety of computational models; namely, algorithms for matrix multiplication, streaming algorithms for the online matrix-vector multiplication pro…

Factoring Polynomials over Finite Fields with Linear Galois Groups: An Additive Combinatorics Approach

Let $\tilde{f}(X)\in\mathbb{Z}[X]$ be a degree-$n$ polynomial such that $f(X):=\tilde{f}(X)\bmod p$ factorizes into $n$ distinct linear factors over $\mathbb{F}_p$. We study the problem of deterministically factoring $f(X)$ over $\mathbb{F}_p$ given $\tilde{f}(X)$. Under the generalized Riemann hypothesis (GRH), we give an improved deterministic algorithm that computes the complete factorization of $f(X)$ in the case that the Galois group of $\tilde{f}(X)$ is (permutation isomorphic to) a linear group $G\leq \mathrm{GL}(V)$ on the set $S$ of roots of $\tilde{f}(X)$, where $V$ is a finite-dimen…

Chang不等式的熵证明

An Entropic Proof of Chang's Inequality

Chang引理是加法组合学和布尔函数分析中的一个有用工具。在这里我们用熵给出一个初等证明。我们得到的常数是紧的，我们给一个轻微的改进的情况下，变量是高度偏置。

The complexity of computing the minimum rank of a sign pattern matrix

We show that computing the minimum rank of a sign pattern matrix is NP hard. Our proof is based on a simple but useful connection between minimum ranks of sign pattern matrices and the stretchability problem for pseudolines arrangements. In fact, our hardness result shows that it is already hard to determine if the minimum rank of a sign pattern matrix is $\leq 3$. We complement this by giving a polynomial time algorithm for determining if a given sign pattern matrix has minimum rank $\leq 2$. Our result answers one of the open problems from Linial et al. [Combinatorica, 27(4):439--463, 2007].

Border rank non-additivity for higher order tensors

Whereas matrix rank is additive under direct sum, in 1981 Schönhage showed that one of its generalizations to the tensor setting, tensor border rank, can be strictly subadditive for tensors of order three. Whether border rank is additive for higher order tensors has remained open. In this work, we settle this problem by providing analogs of Schönhage's construction for tensors of order four and higher. Schönhage's work was motivated by the study of the computational complexity of matrix multiplication; we discuss implications of our results for the asymptotic rank of higher order generalizatio…