research-article

From weak to strong LP gaps for all CSPs

Authors:

Mrinalkanti Ghosh,

Madhur TulsianiAuthors Info & Claims

CCC '17: Proceedings of the 32nd Computational Complexity Conference

Article No.: 11, Pages 1 - 27

Published: 09 July 2017 Publication History

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 [EQUATION] levels of the Sherali-Adams 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 super-constant levels of the Sherali-Adams 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 Sherali-Adams 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 Ω(log log n) levels of the Sherali-Adams hierarchy cannot achieve an approximation better than a random assignment. We simplify their proof and strengthen the bound to [EQUATION] levels.

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Cited By

Cohen-Addad VKarthik CLee EMarx D(2021)On approximability of clustering problems without candidate centersProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458220(2635-2648)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458220
Kothari PManohar P(2021)A stress-free sum-of-squares lower bound for coloringProceedings of the 36th Computational Complexity Conference10.4230/LIPIcs.CCC.2021.23Online publication date: 20-Jul-2021
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2021.23

Index Terms

From weak to strong LP gaps for all CSPs
1. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Other conferences

CCC '17: Proceedings of the 32nd Computational Complexity Conference

July 2017

942 pages

ISBN:9783959770408

Editor:
Ryan O'Donnell
Carnegie Mellon University

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SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

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Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik

Dagstuhl, Germany

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Published: 09 July 2017

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Cited By

Cohen-Addad VKarthik CLee EMarx D(2021)On approximability of clustering problems without candidate centersProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458220(2635-2648)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458220
Kothari PManohar P(2021)A stress-free sum-of-squares lower bound for coloringProceedings of the 36th Computational Complexity Conference10.4230/LIPIcs.CCC.2021.23Online publication date: 20-Jul-2021
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2021.23

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