种植团问题的近紧平方和下界

B. Barak, Samuel B. Hopkins, Jonathan A. Kelner, Pravesh Kothari, Ankur Moitra, Aaron Potechin
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引用次数: 197

摘要

我们证明了从Erdös-Rényi分布G(n,1/2)中选择一个随机图G,对于团问题的n (d)时间阶d平方和半定规划松弛,对于某个常数c > 0,其值至少为n1/2-c(d/log n)1/2,具有高概率。对于任意阶d = o(log n),这产生了该程序值的近紧密的n1/2-o(1)界。此外,我们引入了一个称为伪校准的新框架来构造平方和下界。这个框架的灵感来自于贝叶斯概率论的计算模拟。它给出了构造良好伪分布(即平方和半确定程序的双重证书)的一般方法,并进一步阐明了该层次结构与其他层次结构的不同之处。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem
We prove that with high probability over the choice of a random graph G from the Erdös-Rényi distribution G(n,1/2), the nO(d)-time degree d Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least n1/2-c(d/log n)1/2 for some constant c > 0. This yields a nearly tight n1/2-o(1) bound on the value of this program for any degree d = o(log n). Moreover we introduce a new framework that we call pseudo-calibration to construct Sum-of-Squares lower bounds. This framework is inspired by taking a computational analogue of Bayesian probability theory. It yields a general recipe for constructing good pseudo-distributions (i.e., dual certificates for the Sum-of-Squares semidefinite program), and sheds further light on the ways in which this hierarchy differs from others.
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