Abstract
We study the approximability of constraint satisfaction problems (CSPs) by linear programming (LP) relaxations. We show that for every CSP, the approximation obtained by a basic LP relaxation, is no weaker than the approximation obtained using relaxations given by Omega(log(n)/log(log(n))) levels of the SheraliAdams hierarchy on instances of size n.
It was proved by Chan et al. [FOCS 2013] (and recently strengthened by Kothari et al. [STOC 2017]) that for CSPs, any polynomial size LP extended formulation is no stronger than relaxations obtained by a superconstant levels of the SheraliAdams hierarchy. Combining this with our result also implies that any polynomial size LP extended formulation is no stronger than simply the basic LP, which can be thought of as the base level of the SheraliAdams hierarchy. This essentially gives a dichotomy result for approximation of CSPs by polynomial size LP extended formulations.
Using our techniques, we also simplify and strengthen the result by Khot et al. [STOC 2014] on (strong) approximation resistance for LPs. They provided a necessary and sufficient condition under which Omega(loglog n) levels of the SheraliAdams hierarchy cannot achieve an approximation better than a random assignment. We simplify their proof and strengthen the bound to Omega(log(n)/log(log(n))) levels.
BibTeX  Entry
@InProceedings{ghosh_et_al:LIPIcs:2017:7537,
author = {Mrinalkanti Ghosh and Madhur Tulsiani},
title = {{From Weak to Strong LP Gaps for All CSPs}},
booktitle = {32nd Computational Complexity Conference (CCC 2017)},
pages = {11:111:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770408},
ISSN = {18688969},
year = {2017},
volume = {79},
editor = {Ryan O'Donnell},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7537},
URN = {urn:nbn:de:0030drops75370},
doi = {10.4230/LIPIcs.CCC.2017.11},
annote = {Keywords: Constraint Satisfaction Problem, Convex Programming, Linear Programming Hierarchy, Integrality Gap}
}
Keywords: 

Constraint Satisfaction Problem, Convex Programming, Linear Programming Hierarchy, Integrality Gap 
Seminar: 

32nd Computational Complexity Conference (CCC 2017) 
Issue Date: 

2017 
Date of publication: 

21.07.2017 