Hyperstable Sets with Voting and Algorithmic Hardness Applications

TL;DR

This paper introduces the concept of hyperstable partitions in Gaussian space, classifies their properties, and explores their implications for conjectures and hardness results in computational complexity.

Contribution

It defines hyperstability for partitions, links it to key conjectures, and suggests that proving hyperstability could resolve major open problems in noise stability and hardness.

Findings

Hyperstable partitions are critical points for noise stability derivatives.

Symmetric hyperstable sets must be star-shaped.

Hyperstability implies equal measure parts in partitions.

Abstract

The noise stability of a Euclidean set $A$ with correlation $\rho$ is the probability that $(X,Y)\in A\times A$, where $X,Y$ are standard Gaussian random vectors with correlation $\rho\in(0,1)$. It is well-known that a Euclidean set of fixed Gaussian volume that maximizes noise stability must be a half space. For a partition of Euclidean space into $m>2$ parts each of Gaussian measure $1/m$, it is still unknown what sets maximize the sum of their noise stabilities. In this work, we classify partitions maximizing noise stability that are also critical points for the derivative of noise stability with respect to $\rho$. We call a partition satisfying these conditions hyperstable. Uner the assumption that a maximizing partition is hyperstable, we prove: * a (conditional) version of the Plurality is Stablest Conjecture for $3$ or $4$ candidates. * a (conditional) sharp Unique Games Hardness result for MAX-m-CUT for $m=3$ or $4$ * a (conditional) version of the Propeller Conjecture of Khot and Naor for $4$ sets. We also show that a symmetric set that is hyperstable must be star-shaped. For partitions of Euclidean space into $m>2$ parts of fixed (but perhaps unequal) Gaussian measure, the hyperstable property can only be satisfied when all of the parts have Gaussian measure $1/m$. So, as our main contribution, we have identified a possible strategy for proving the full Plurality is Stablest Conjecture and the full sharp hardness for MAX-m-CUT: to prove both statements, it suffices to show that sets maximizing noise stability are hyperstable. This last point is crucial since any proof of the Plurality is Stablest Conjecture must use a property that is special to partitions of sets into equal measures, since the conjecture is false in the unequal measure case.