We examine the tradeoff between privacy and usability of statistical databases. We model a statistical database by an n-bit string d1, .., dn, with a query being a subset q ⊆ [n] to be answered by i∈q di. Our main result is a polynomial reconstruction algorithm of data from noisy (perturbed) subset sums. Applying this reconstruction algorithm to statistical databases we show that in order to achieve privacy one has to add perturbation of magnitude Ω( √ n). That is, smaller perturbation always results in a strong violation of privacy. We show that this result is tight by exemplifying access algorithms for statistical databases that preserve privacy while adding perturbation of magnitudeÕ( √ n). For time-T bounded adversaries we demonstrate a privacypreserving access algorithm whose perturbation magnitude is ≈ √ T .
We present a new proof of the PCP theorem that is based on a combinatorial amplification lemma. The unsat value of a set of constraints C = {c1, . . . , cn}, denoted UNSAT(C), is the smallest fraction of unsatisfied constraints, ranging over all possible assignments for the underlying variables.We describe a new combinatorial amplification transformation that doubles the unsat-value of a constraint-system, with only a linear blowup in the size of the system. The amplification step causes an increase in alphabet-size that is corrected by a PCP composition step. Iterative application of these two steps yields a proof for the PCP theorem.The amplification lemma relies on a new notion of "graph powering" that can be applied to systems of constraints. This powering amplifies the unsat-value of a constraint system provided that the underlying graph structure is an expander.We also apply the amplification lemma to construct PCPs and locally-testable codes whose length is linear up to a polylog factor, and whose correctness can be probabilistically verified by making a constant number of queries. Namely, we prove SAT ∈ P CP 1 2 ,1 [log 2 (n · poly log n), O(1)].
We prove the Minimum Vertex Cover problem to be NP-hard to approximate to within a factor of 1.3606, extending on previous PCP and hardness of approximation technique. To that end, one needs to develop a new proof framework, and to borrow and extend ideas from several fields.
We propose an analytical framework for studying parallel repetition, a basic product operation for one-round twoplayer games. In this framework, we consider a relaxation of the value of projection games. We show that this relaxation is multiplicative with respect to parallel repetition and that it provides a good approximation to the game value. Based on this relaxation, we prove the following improved parallel repetition bound: For every projection game G with value at most ρ, the k-fold parallel repetition G ⊗k has value at most val(G ⊗k )This statement implies a parallel repetition bound for projection games with low value ρ. Previously, it was not known whether parallel repetition decreases the value of such games. This result allows us to show that approximating set cover to within factor (1 − ε) ln n is NP-hard for every ε > 0, strengthening Feige's quasi-NP-hardness and also building on previous work by Moshkovitz and Raz.In this framework, we also show improved bounds for few parallel repetitions of projection games, showing that Raz's counterexample to strong parallel repetition is tight even for a small number of repetitions.Finally, we also give a short proof for the NP-hardness of label cover(1, δ) for all δ > 0, starting from the basic PCP theorem.
PTSD symptoms predicted poor disease control in this cohort of MI survivors. The data suggest that screening MI survivors for symptoms of PTSD may be beneficial if this high-risk population is to be targeted for interventions.
We describe a new parameterized family of symmetric error-correcting codes with low-density parity-check matrices (LDPC).Our codes can be described in two seemingly different ways. First, in relation to Reed-Muller codes: our codes are functions on a subset of F n whose restrictions to a prescribed set of affine lines has low degree. Alternatively, they are Tanner codes on high dimensional expanders, where the coordinates of the codeword correspond to triangles of a 2-dimensional expander, such that around every edge the local view forms a Reed-Solomon codeword.For some range of parameters our codes are provably locally testable, and their dimension is some fixed power of the block length. For another range of parameters our codes have distance and dimension that are both linear in the block length, but we do not know if they are locally testable. The codes also have the multiplication property: the coordinate-wise product of two codewords is a codeword in a related code.The definition of the codes relies on the construction of a specific family of simplicial complexes which is a slight variant on the coset complexes of Kaufman and Oppenheim. We show a novel way to embed the triangles of these complexes into F n , with the property that links of edges embed as affine lines in F n .We rely on this embedding to lower bound the rate of these codes in a way that avoids constraint-counting and thereby achieves non-trivial rate even when the local codes themselves have arbitrarily small rate, and in particular below 1/2.
Given a k-uniform hyper-graph, the Ek-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyper-edge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that Ek-Vertex-Cover is NP-hard to approximate within factor (k − 1 − ε) for any k ≥ 3 and any ε > 0. The result is essentially tight as this problem can be easily approximated within factor k. Our construction makes use of the biased Long-Code and is analyzed using combinatorial properties of s-wise t-intersecting families of subsets.
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