线性规划的逼近极限（超越层次）
Approximation Limits of Linear Programs (Beyond Hierarchies)
摘要：
我们开发了一个框架的多项式规模的线性规划的近似极限从下界的非负秩适当定义的矩阵。这个框架产生无条件的不可能性的结果，适用于任何线性规划，而不是只有程序生成的层次结构。使用我们的框架，我们证明了CLIQUE的O（n^{1/2-eps}）-近似需要大小为2^{n^Ω（eps）}的线性规划。(This下限适用于使用CLIQUE的特定编码作为线性优化问题的线性程序。此外，我们建立了半定规划的线性规划的近似类似的结果。我们的主要成分是一个定量的改进Razborov的矩形腐败引理的高误差制度，它给出了强有力的下界的非负秩的某些扰动的独特的不相交矩阵。
英文摘要：
We develop a framework for approximation limits of polynomial-size linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n^{1/2-eps})-approximations for CLIQUE require linear programs of size 2^{n^Ω(eps)}. (This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem.) Moreover, we establish a similar result for approximations of semidefinite programs by linear programs. Our main ingredient is a quantitative improvement of Razborov's rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of certain perturbations of the unique disjointness matrix.
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