Approximation Limits of Linear Programs (Beyond Hierarchies)

We develop a framework for proving 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 quadratic approximations for CLIQUE require linear programs of exponential size. (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 semi definite programs by linear programs. Our main technical ingredient is a quantitative improvement of Razborov's rectangle corruption lemma (1992) for the high error regime, which gives strong lower bounds on the nonnegative rank of certain perturbations of the unique disjoint ness matrix.