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.
We prove a quantitative version of the Gibbard-Satterthwaite theorem. We show that a uniformly chosen voter profile for a neutral social choice function f of q ≥ 4 alternatives and n voters will be manipulable with probability at least 10 −4 2 n −3 q −30 , where is the minimal statistical distance between f and the family of dictator functions.Our results extend those of [11], which were obtained for the case of 3 alternatives, and imply that the approach of masking manipulations behind computational hardness (as considered in [4,6,9,15,7]) cannot hide manipulations completely.Our proof is geometric. More specifically it extends the method of canonical paths to show that the measure of the profiles that lie on the interface of 3 or more outcomes is large. To the best of our knowledge our result is the first isoperimetric result to establish interface of more than two bodies.
We prove a quantitative version of the Gibbard-Satterthwaite theorem. We show that a uniformly chosen voter profile for a neutral social choice function f of q ≥ 4 alternatives and n voters will be manipulable with probability at least 10 −4 ǫ 2 n −3 q −30 , where ǫ is the minimal statistical distance between f and the family of dictator functions.Our results extend those of [FKN09], which were obtained for the case of 3 alternatives, and imply that the approach of masking manipulations behind computational hardness (as considered in [BO91, CS03, EL05, PR06, CS06]) cannot hide manipulations completely.Our proof is geometric. More specifically it extends the method of canonical paths to show that the measure of the profiles that lie on the interface of 3 or more outcomes is large. To the best of our knowledge our result is the first isoperimetric result to establish interface of more than two bodies.
We study constraint satisfaction problems on the domain {−1, 1}, where the given constraints are homogeneous linear threshold predicates, that is, predicates of the form sgn(w 1 x 1 + · · · + w n x n ) for some positive integer weights w 1 , . . . , w n . Despite their simplicity, current techniques fall short of providing a classification of these predicates in terms of approximability. In fact, it is not easy to guess whether there exists a homogeneous linear threshold predicate that is approximation resistant or not.The focus of this article is to identify and study the approximation curve of a class of threshold predicates that allow for nontrivial approximation. Arguably the simplest such predicate is the majority predicate sgn(x 1 + · · · + x n ), for which we obtain an almost complete understanding of the asymptotic approximation curve, assuming the Unique Games Conjecture. Our techniques extend to a more general class of "majoritylike" predicates and we obtain parallel results for them. In order to classify these predicates, we introduce the notion of Chow-robustness that might be of independent interest.
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