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Volume 12 (2016) Article 4 pp. 1-50
Majority is Stablest: Discrete and SoS
Received: September 11, 2013
Revised: September 9, 2014
Published: July 17, 2016
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Keywords: Majority is stablest, Sum-of-Squares hierarchy, Gaussian isoperimetry
Categories: complexity theory, Boolean functions, majority, sum-of-squares hierarchy, isoperimetry, Gaussian isoperimetry
ACM Classification: F.2, G.3
AMS Classification: 68Q17, 60G15, 60D99

Abstract: [Plain Text Version]

The “Majority is Stablest” Theorem has numerous applications in hardness of approximation and social choice theory. We give a new proof of the “Majority is Stablest” Theorem by induction on the dimension of the discrete cube. Unlike the previous proof, it uses neither the “invariance principle” nor Borell's result in Gaussian space. Moreover, the new proof allows us to derive a proof of “Majority is Stablest” in a constant level of the Sum of Squares hierarchy. This implies in particular that the Khot-Vishnoi instance of Max-Cut does not provide a gap instance for the Lasserre hierarchy.

An extended abstract of this paper appeared in the Proceedings of the 45th ACM Symposium on the Theory of Computing, 2013.