Machinery for Proving Sum-of-Squares Lower Bounds on Certification Problems

行列式对称电路的下界

Lower Bounds for Symmetric Circuits for the Determinant

本文针对行列式计算提出更严格的对称算术电路要求，基于支撑定理和双人受限双射博弈开发了下界证明框架，并证明在此对称性下计算行列式存在指数级下界。

Sum-of-Squares Lower Bounds for Sherrington-Kirkpatrick via Planted Affine Planes

The Sum-of-Squares (SoS) hierarchy is a semi-definite programming meta-algorithm that captures state-of-the-art polynomial time guarantees for many optimization problems such as Max-$k$-CSPs and Tensor PCA. On the flip side, a SoS lower bound provides evidence of hardness, which is particularly relevant to average-case problems for which NP-hardness may not be available. In this paper, we consider the following average case problem, which we call the \emph{Planted Affine Planes} (PAP) problem: Given $m$ random vectors $d_1,\ldots,d_m$ in $\mathbb{R}^n$, can we prove that there is no vector $v …

New lower bounds for Massively Parallel Computation from query complexity

Roughgarden, Vassilvitskii, and Wang (JACM 18) recently introduced a novel framework for proving lower bounds for Massively Parallel Computation using techniques from boolean function complexity. We extend their framework in two different ways, to capture two common features of Massively Parallel Computation: $\circ$ Adaptivity, where machines can write to and adaptively read from shared memory throughout the execution of the computation. Recent work of Behnezhad et al. (SPAA 19) showed that adaptivity enables significantly improved round complexities for a number of central graph problems. $\…

Time-Space Tradeoffs for Distinguishing Distributions and Applications to Security of Goldreich's PRG

In this work, we establish lower-bounds against memory bounded algorithms for distinguishing between natural pairs of related distributions from samples that arrive in a streaming setting. In our first result, we show that any algorithm that distinguishes between uniform distribution on $\{0,1\}^n$ and uniform distribution on an $n/2$-dimensional linear subspace of $\{0,1\}^n$ with non-negligible advantage needs $2^{Ω(n)}$ samples or $Ω(n^2)$ memory. Our second result applies to distinguishing outputs of Goldreich's local pseudorandom generator from the uniform distribution on the output domai…

一个动态布尔数据结构的细胞探测下界：$Ω((\log n / \log \log n)^2)$

An $Ω((\log n/\log\log n)^2)$ Cell-Probe Lower Bound for Dynamic Boolean Data Structures

本文解决了动态布尔数据结构的长期开放问题，证明了Patrascu的多阶段问题下界，展示了将 discrepancy 下界转化为动态布尔数据结构下界的一般方法。