Approximation Limits of Linear Programs (Beyond Hierarchies)

TL;DR

This paper introduces a new framework for establishing approximation limits of polynomial-size linear programs, providing strong lower bounds that apply broadly beyond hierarchical methods.

Contribution

The authors develop a general approach based on nonnegative rank bounds to prove approximation impossibility results for linear and semidefinite programs.

Findings

Linear programs cannot efficiently approximate CLIQUE within certain bounds.

Established lower bounds on nonnegative rank for specific matrix perturbations.

Extended results to semidefinite program approximations by linear programs.

Abstract

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^\Omega(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.