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Approximation Limits of Linear Programs (Beyond Hierarchies) (2013)
| Citations: | 17 - 6 self |
BibTeX
@MISC{Braun13approximationlimits,
author = {Gábor Braun and Samuel Fiorini and Sebastian Pokutta and David Steurer},
title = {Approximation Limits of Linear Programs (Beyond Hierarchies)},
year = {2013}
}
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Abstract
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 O(n1/2−ɛ)-approximations for CLIQUE require linear programs of size 2nΩ(ɛ). 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 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 shifts of the unique disjointness matrix.
Keyphrases
linear program approximation limit beyond hierarchy nonnegative rank semidefinite program quantitative improvement unconditional impossibility result unique disjointness matrix rectangle corruption lemma similar result polynomial-size linear program bound applies high error regime linear optimization problem main technical ingredient certain encoding
