Edge-isoperimetric inequalities and ball-noise stability: Linear programming and probabilistic approaches

Abstract

Let Qnr be the graph with vertex set {−1,1}n in which two vertices are joined if their Hamming distance is at most r. The edge-isoperimetric problem for Qnr is that: For every (n,r,M) such that 1≤r≤n and 1≤M≤2n, determine the minimum edge-boundary size of a subset of vertices of Qnr with a given size M. In this paper, we apply two different approaches to prove bounds for this problem. The first approach is a linear programming approach and the second is probabilistic. Our bound derived by the first approach generalizes the tight bound for M=2n−1 derived by Kahn, Kalai, and Linial in 1989. Moreover, our bound is also tight for M=2n−2 and r≤n2−1. Our bounds derived by the second approach are expressed in terms of the noise stability, and they are shown to be asymptotically tight as n→∞ when r=2⌊βn2⌋+1 and M=⌊α2n⌋ for fixed α,β∈(0,1), and is tight up to a factor 2 when r=2⌊βn2⌋ and M=⌊α2n⌋. In fact, the edge-isoperimetric problem is equivalent to a ball-noise stability problem which is a variant of the traditional (i.i.d.-) noise stability problem. Our results can be interpreted as bounds for the ball-noise stability problem.