Applications of Gaussian Noise Stability in Inapproximability and Social Choice Theory

Licentiate thesis, 2009

Gaussian isoperimetric results have recently played an important role in proving fundamental results in hardness of approximation in computer science and in the study of voting schemes in social choice theory. In this thesis we prove a generalization of a Gaussian isoperimetric result by Borell and show that it implies that the majority function is optimal in Condorcet voting in the sense that it maximizes the probability that there is a single candidate which the society prefers over all other candidates. We also show that a different Gaussian isoperimetric conjecture which can be viewed as a generalization of the ''Double Bubble'' theorem implies the Plurality is Stablest conjecture and also that the Frieze-Jerrum semidefinite programming based algorithm for MAX-q-CUT achieves the optimal approximation factor assuming the Unique Games Conjecture. Both applications crucially depend on the invariance principle of Mossel, O'Donnell and Oleszkiewicz which lets us rephrase questions about noise stability of low-influential discrete functions in terms of noise stability of functions on R^n under Gaussian measure. We prove a generalization of this invariance principle needed for our applications.