Social choice, computational complexity, Gaussian geometry, and Boolean functions
Abstract
We describe a web of connections between the following topics: the mathematical theory of voting and social choice; the computational complexity of the Maximum Cut problem; the Gaussian Isoperimetric Inequality and Borell's generalization thereof; the Hypercontractive Inequality of Bonami; and, the analysis of Boolean functions. A major theme is the technique of reducing inequalities about Gaussian functions to inequalities about Boolean functions f : {-1,1}^n -> {-1,1}, and then using induction on n to further reduce to inequalities about functions f : {-1,1} -> {-1,1}. We especially highlight De, Mossel, and Neeman's recent use of this technique to prove the Majority Is Stablest Theorem and Borell's Isoperimetric Inequality simultaneously.
Cite
@article{arxiv.1407.7763,
title = {Social choice, computational complexity, Gaussian geometry, and Boolean functions},
author = {Ryan O'Donnell},
journal= {arXiv preprint arXiv:1407.7763},
year = {2014}
}
Comments
In proceedings of the 2014 ICM. Corrected a few minor typos from previous version