Maximally Stable Gaussian Partitions with Discrete Applications

TL;DR

This paper advances Gaussian noise stability theory, generalizing Borell's isoperimetric result, and applies it to voting, approximation algorithms, and conjectures in social choice and theoretical computer science.

Contribution

It proves a new Gaussian noise stability result that generalizes Borell's isoperimetric theorem and connects to multiple conjectures and optimality results in social choice and approximation algorithms.

Findings

Proved a generalized Gaussian noise stability theorem.

Established optimality of majority in Condorcet voting.

Connected noise stability conjecture to the Plurality is Stablest Conjecture.

Abstract

Gaussian noise stability results have recently played an important role in proving results in hardness of approximation in computer science and in the study of voting schemes in social choice. We prove a new Gaussian noise stability result generalizing an isoperimetric result by Borell on the heat kernel and derive as applications: * An optimality result for majority in the context of Condorcet voting. * A proof of a conjecture on "cosmic coin tossing" for low influence functions. We also discuss a Gaussian noise stability conjecture which may be viewed as a generalization of the "Double Bubble" theorem and show that it implies: * A proof of the "Plurality is Stablest Conjecture". * That the Frieze-Jerrum SDP for MAX-q-CUT achieves the optimal approximation factor assuming the Unique Games Conjecture.