Approximation Limits of Linear Programs (Beyond Hierarchies)

We develop a framework for proving approximation limits of polynomial size linear programs (LPs) from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any LP as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n^1/2-ϵ)-approximations for CLIQUE require LPs of size $2^{n^{\text{Ω}(ϵ)}}$. This lower bound applies to LPs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by LPs.

Our main technical ingredient is a quantitative improvement of Razborov’s [38] rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix.