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We prove the following strong hardness result for learning: Given a distribution on labeled examples from the hypercube such that there exists a monomial (or conjunction) consistent with (1-ε)-fraction of the examples, it is NP-hard to find a halfspace that is correct on ( 1/2 +ε)-fraction of the examples, for arbitrary constant ε(More)
In this paper we study a fundamental open problem in the area of probabilistic checkable proofs: What is the smallest s such that NP ⊆ naPCP1,s[O(log n),3]? In the language of hardness of approximation, this problem is equivalent to determining the smallest s such that getting an s-approximation for satisfiable 3-bit constraint satisfaction problems(More)
We study the approximability of multiway partitioning problems, examples of which include Multiway Cut, Node-weighted Multiway Cut, and Hypergraph Multiway Cut. We investigate these problems from the point of view of two possible generalizations: as Min-CSPs, and as Submodular Multiway Partition problems. These two generalizations lead to two natural(More)
We study lower bounds for Locality-Sensitive Hashing (LSH) in the strongest setting: point sets in {0,1}<sup>d</sup> under the Hamming distance. Recall that H is said to be an (<i>r</i>, <i>cr</i>, <i>p</i>, <i>q</i>)-sensitive hash family if all pairs <i>x</i>, <i>y</i> &#8712; {0,1}<sup>d</sup> with dist(<i>x</i>, <i>y</i>) &#8804; <i>r</i> have(More)
We construct pseudorandom generators that fool functions of halfspaces (threshold functions) under a very broad class of product distributions. This class includes not only familiar cases such as the uniform distribution on the discrete cube, the uniform distribution on the solid cube, and the multivariate Gaussian distribution, but also includes any(More)
We introduce a novel privacy framework that we call Membership Privacy. The framework includes positive membership privacy, which prevents the adversary from significantly increasing its ability to conclude that an entity is in the input dataset, and negative membership privacy, which prevents leaking of non-membership. These notions are parameterized by a(More)
Let G be an undirected graph for which the standard Max-Cut SDP relaxation achieves at least a c fraction of the total edge weight, 1/2 <= c <= 1. If the actual optimal cut for G is at most an s fraction of the total edge weight, we say that (c, s) is an SDP gap. We define the SDP gap curve GapSDP : [1/2,1] -&#62; [1/2,1] by GapSDP(c) = inf{s : (c, s) is an(More)
The Unique Games Conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for <i>none</i> of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance nevertheless seems critical in these proofs. In this work, we bypass the(More)
In the conclusion of his monumental paper on optimal inapproximability results, Håstad [13] suggested that Fourier analysis of Dictator (Long Code) Tests may not be universally applicable in the study of CSPs. His main open question was to determine if the technique could resolve the approximability of satisfiable 3-bit constraint satisfaction problems. In(More)