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Optimal algorithms and inapproximability results for every CSP
 In Proc. 40 th ACM STOC
, 2008
"... Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the ..."
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Cited by 144 (13 self)
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Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the following result: If UGC is true, then for every constraint satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a generic conversion from SDP integrality gaps to UGC hardness results for every CSP. This result holds both for maximization and minimization problems over arbitrary finite domains. Using this connection between integrality gaps and hardness results we obtain a generic polynomialtime algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every CSP. Unconditionally, for all 2CSPs the algorithm achieves an approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut
A generalization of the Lindeberg principle
 Annals Probab
, 2006
"... We present a generalization of Lindeberg’s method of proving the central limit theorem to encompass general smooth functions (instead of just sums) and dependent random variables. The technique is then used to obtain an invariance result for smooth functions of exchangeable random variables. As an i ..."
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Cited by 43 (1 self)
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We present a generalization of Lindeberg’s method of proving the central limit theorem to encompass general smooth functions (instead of just sums) and dependent random variables. The technique is then used to obtain an invariance result for smooth functions of exchangeable random variables. As an illustrative application of this theorem, we then establish “convergence to Wigner’s law ” for eigenspectra of matrices with exchangeable random entries. 1 Introduction and
Agnostic Learning of Monomials by Halfspaces is Hard
"... Abstract — 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 NPhard to find a halfspace that is correct on ( 1 +ϵ)fractio ..."
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Cited by 26 (10 self)
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Abstract — 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 NPhard to find a halfspace that is correct on ( 1 +ϵ)fraction of the examples, 2 for arbitrary constant ϵ> 0. In learning theory terms, weak agnostic learning of monomials by halfspaces is NPhard. This hardness result bridges between and subsumes two previous results which showed similar hardness results for the proper learning of monomials and halfspaces. As immediate corollaries of our result, we give the first optimal hardness results for weak agnostic learning of decision lists and majorities. Our techniques are quite different from previous hardness proofs for learning. We use an invariance principle and sparse approximation of halfspaces from recent work on fooling halfspaces to give a new natural list decoding of a halfspace in the context of dictatorship tests/label cover reductions. In addition, unlike previous invariance principle based proofs which are only known to give Unique Games hardness, we give a reduction from a smooth version of Label Cover that is known to be NPhard.
On the unique games conjecture
 In FOCS
, 2005
"... This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1 ..."
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Cited by 15 (1 self)
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This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1
Inapproximability of NPcomplete problems, discrete fourier analysis, and geometry
 In Proc. the International Congress of Mathematicians
, 2004
"... Abstract. This article gives a survey of recent results that connect three areas in computer science and mathematics: (1) (Hardness of) computing approximate solutions to NPcomplete problems. (2) Fourier analysis of boolean functions on boolean hypercube. (3) Certain problems in geometry, especiall ..."
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Cited by 7 (2 self)
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Abstract. This article gives a survey of recent results that connect three areas in computer science and mathematics: (1) (Hardness of) computing approximate solutions to NPcomplete problems. (2) Fourier analysis of boolean functions on boolean hypercube. (3) Certain problems in geometry, especially related to isoperimetry and embeddings between metric spaces.
Limiting spectral distribution for wigner matrices with dependent entries. arXiv preprint arXiv:1304.3394
, 2013
"... ar ..."
The Approximability of Learning and Constraint Satisfaction Problems
, 2010
"... International Business Machine. The views and conclusions contained in this document are those of the ..."
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International Business Machine. The views and conclusions contained in this document are those of the