In this paper we study functions with low influences on product probability spaces. These are functions f W 1 n ! ‫ޒ‬ that have EOEVar i OEf small compared to VarOEf for each i . The analysis of boolean functions f W f 1; 1g n ! f 1; 1g with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics.We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known nonlinear invariance principles. It has the advantage that its proof is simple and that its error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly "smoothed"; this extension is essential for our applications to "noise stability"-type problems.In particular, as applications of the invariance principle we prove two conjectures: Khot, Kindler, Mossel, and O'Donnell's "Majority Is Stablest" conjecture from theoretical computer science, which was the original motivation for this work, and Kalai and Friedgut's "It Ain't Over Till It's Over" conjecture from social choice theory.

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of α GW + ∈, for all ∈ > 0; here α GW ≈ .878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [26]. This implies that if the Unique Games Conjecture of Khot [37] holds then the Goemans-Williamson approximation algorithm is optimal. Our result indicates that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem. Our reduction relies on a theorem we call Majority Is Stablest. This was introduced as a conjecture in the original version of this paper, and was subsequently confirmed in [45]. A stronger version of this conjecture called Plurality Is Stablest is still open, although [45] contains a proof of an asymptotic version of it. Our techniques extend to several other two-variable constraint satisfaction problems. In particular, subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX-2SAT, MAX-q-CUT, and MAX-2LIN(q). For MAX-2SAT we show approximation hardness up to a factor of roughly .943. This nearly matches the .940 approximation algorithm of Lewin, Livnat, and Zwick [41]. Furthermore, we show that our .943... factor is actually tight for a slightly restricted version of MAX-2SAT. For MAX-q-CUT we show a hardness factor which asymptotically (for large q) matches the approximation factor achieved by Frieze and Jerrum [25], namely 1 − 1/q + 2(ln q)/q 2. For MAX-2LIN(q) we show hardness of distinguishing between instances which are (1−∈)-satisfiable and those which are not even, roughly, (q −∈/2)-satisfiable. These parameters almost match those achieved by the recent algorithm of Charikar, Makarychev, and Makarychev [10]. The hardness result holds even for instances in which all equations are of the form x i − x j = c. At a more qualitative level, this result also implies that 1 − ∈ vs. ∈ hardness for MAX-2LIN(q) is equivalent to the Unique Games Conjecture.

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of α GW + ∈, for all ∈ > 0; here α GW ≈ .878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [26]. This implies that if the Unique Games Conjecture of Khot [37] holds then the Goemans-Williamson approximation algorithm is optimal. Our result indicates that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem.Our reduction relies on a theorem we call Majority Is Stablest. This was introduced as a conjecture in the original version of this paper, and was subsequently confirmed in [45]. A stronger version of this conjecture called Plurality Is Stablest is still open, although [45] contains a proof of an asymptotic version of it.Our techniques extend to several other two-variable constraint satisfaction problems. In particular, subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX-2SAT, MAX-q-CUT, and MAX-2LIN(q).For MAX-2SAT we show approximation hardness up to a factor of roughly .943. This nearly matches the .940 approximation algorithm of Lewin, Livnat, and Zwick [41]. Furthermore, we show that our .943... factor is actually tight for a slightly restricted version of MAX-2SAT. For MAX-q-CUT we show a hardness factor which asymptotically (for large q) matches the approximation factor achieved by Frieze and Jerrum [25], namely 1 − 1/q + 2(ln q)/q 2 .For MAX-2LIN(q) we show hardness of distinguishing between instances which are (1−∈)-satisfiable and those which are not even, roughly, (q −∈/2 )-satisfiable. These parameters almost match those achieved by the recent algorithm of Charikar, Makarychev, and Makarychev [10]. The hardness result holds even for instances in which all equations are of the form x i − x j = c. At a more qualitative level, this result also implies that 1 − ∈ vs. ∈ hardness for MAX-2LIN(q) is equivalent to the Unique Games Conjecture. AbstractIn this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of α GW + , for all > 0; here α GW ≈ .878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [26]. This implies that if the Unique Games Conjecture of Khot [37] holds then the Goemans-Williamson approximation algorithm is optimal. Our result indicates that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem.Our reduction relies on a theorem we call Majority Is Stablest. This was introduced as a conjecture in the original version of this paper, and was subsequently confirmed in [45]. A stronger version of this conjecture called Plurality Is Stablest is still open, although [45] contains a proof of an asymptotic version of it.Our techniques extend to several other two-variable constraint satisfaction problems. In particular, subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX-2SAT, M...

In the (deletion-channel) trace reconstruction problem, there is an unknown n-bit source string x. An algorithm is given access to independent traces of x, where a trace is formed by deleting each bit of x independently with probability δ. The goal of the algorithm is to recover x exactly (with high probability), while minimizing samples (number of traces) and running time. Previously, the best known algorithm for the trace reconstruction problem was due to Holenstein et al. [SODA 2008]; it uses exp(O (n 1/2)) samples and running time for any xed 0 < δ < 1. It is also what we call a "mean-based algorithm", meaning that it only uses the empirical means of the individual bits of the traces. Holenstein et al. also gave a lower bound, showing that any mean-based algorithm must use at least n Ω(log n) samples. In this paper we improve both of these results, obtaining matching upper and lower bounds for mean-based trace reconstruction. For any constant deletion rate 0 < δ < 1, we give a mean-based algorithm that uses exp(O (n 1/3)) time and traces; we also prove that any mean-based algorithm must use at least exp(Ω(n 1/3)) traces. In fact, we obtain matching upper and lower bounds even for δ subconstant and ρ 1 − δ subconstant: when (log 3 n)/n δ ≤ 1/2 the bound is exp(−Θ(δn) 1/3), and when 1/ √ n ρ ≤ 1/2 the bound is exp(−Θ(n/ρ) 1/3). Our proofs involve estimates for the maxima of Littlewood polynomials on complex disks. We show that these techniques can also be used to perform trace reconstruction with random insertions and bit-ips in addition to deletions. We also nd a surprising result: for deletion probabilities δ > 1/2, the presence of insertions can actually help with trace reconstruction.

We study the learnability of sets in R n under the Gaussian distribution, taking Gaussian surface area as the "complexity measure" of the sets being learned. Let C S denote the class of all (measurable) sets with surface area at most S. We first show that the class C S is learnable to any constant accuracy in time n O(S 2 ) , even in the arbitrary noise ("agnostic") model. Complementing this, we also show that any learning algorithm for C S information-theoretically requires 2 Ω(S 2 ) examples for learning to constant accuracy. These results together show that Gaussian surface area essentially characterizes the computational complexity of learning under the Gaussian distribution.Our approach yields several new learning results, including the following (all bounds are for learning to any constant accuracy):• The class of all convex sets can be agnostically learned in time 2Õ( √ n) (and we prove a 2lower bound for noise-free learning). This is the first subexponential time algorithm for learning general convex sets even in the noise-free (PAC) model.