SDP Integrality Gaps with Local ell_1-Embeddability

Khot, Subhash; Saket, Rishi

doi:10.1109/focs.2009.37

Cited by 45 publications

(49 citation statements)

References 14 publications

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“…We propose semidefinite programming (SDP) hierarchies as a natural and powerful class of algorithmic attacks to rule out. Briefly, we will be considering the "basic" SDP hierarchy known as SheraliAdams + (SA + ) [13], [14], [15], as well as the extremely powerful Lasserre/Parrilo/Sum-of-Squares SDP hierarchy [16], [17]. (For more details, see Section II.)…”

Section: A Our Resultsmentioning

confidence: 99%

Goldreich's PRG: Evidence for Near-Optimal Polynomial Stretch

O’Donnell

Witmer

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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Abstract-Furthering the study of cryptography in NC 0 , we give new evidence for the security of Goldreich's candidate pseudorandom generator with near-optimal, polynomial stretch. Our evidence consists both of security against subexponential-time F2-linear attacks as well as subexponential-time attacks using SDP hierarchies such as Sherali-Adams + and Lasserre/Parrilo. More specifically, instantiating Goldreich's generator with the predicate P (x1, . . . , x5) = x1+x2+x3+x4x5 (mod 2) we get a candidate 5-local PRG which stretches n bits to n 1.499 bits and which is secure against both F2-linear attacks and attacks based on the Lasserre/Parrilo SDP hierarchy. Previous works with such small locality only gave stretch n 1.249 and were only shown to be secure against F2-linear attacks.Our result is essentially optimal, as known SDP/spectral techniques show the generator would not be secure if used with stretch Θ(n 3/2 log n). More generally, when (a slight variant of) Goldreich's generator is used with a local predicate P (x) which is (t− 1)-wise independent, we show that one can allow stretch n t/2− for any > 0 while resisting subexponentialtime attacks based on the Sherali-Adams + SDP hierarchy. Again, this is amount of stretch is (potentially) optimal due to known SDP/spectral methods which succeed at stretch Θ(n t/2 log n). Finally, for a large family of predicates we also extend this result to security against the much stronger Lasserre/Parrilo SDP hierarchy.

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Section: A Our Resultsmentioning

confidence: 99%

Goldreich's PRG: Evidence for Near-Optimal Polynomial Stretch

O’Donnell

Witmer

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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“…Using an alternate construction based on the geometry of Heisenberg group, a sequence of works by Lee and Naor [36], Cheeger and Kleiner [11,12], Cheeger, Kleiner, and Naor [13,14] The lower bound of Theorem 5.3 is also strengthened in a different direction by Raghavendra and Steurer [40] (also by Khot and Saket [33] with quantitatively weaker result): Theorem 5.6. ( [40,33]) There is an N -point negative type metric such that its submetric on any subset of t points is isometrically 1 -embeddable, but the whole metric incurs distortion of at least t to embed into 1 , and t = (log log N ) c for some constant c > 0.…”

Section: Connection To Metric Embeddingsmentioning

confidence: 99%

Inapproximability of NP-complete Problems, Discrete Fourier Analysis, and Geometry

Khot

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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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 NP-complete problems. (2) Fourier analysis of boolean functions on boolean hypercube. (3) Certain problems in geometry, especially related to isoperimetry and embeddings between metric spaces. Mathematics Subject Classification (2000). Primary 68Q17.

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“…Indeed the approach via Unique Games remains the only known way to get such strong gaps for Max Cut. Recently, even stronger gaps for Max-Cut were shown using this framework in [9,13]. Another example of a basic problem for which a SDP gap construction is only known via the reduction from Unique Games is Maximum Acyclic Subgraph [5].…”

Section: Sdp Gaps As a Reduction Toolmentioning

confidence: 99%

SDP Gaps for 2-to-1 and Other Label-Cover Variants

Guruswami

Khot

O’Donnell

et al. 2010

Automata, Languages and Programming

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Abstract. In this paper we present semidefinite programming (SDP) gap instances for the following variants of the Label-Cover problem, closely related to the Unique Games Conjecture: (i) 2-to-1 Label-Cover; (ii) 2-to-2 Label-Cover; (iii) α-constraint Label-Cover. All of our gap instances have perfect SDP solutions. For alphabet size K, the integral optimal solutions have value:Prior to this work, there were no known SDP gap instances for any of these problems with perfect SDP value and integral optimum tending to 0.

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SDP Integrality Gaps with Local ell_1-Embeddability

Cited by 45 publications

References 14 publications

Goldreich's PRG: Evidence for Near-Optimal Polynomial Stretch

Goldreich's PRG: Evidence for Near-Optimal Polynomial Stretch

Inapproximability of NP-complete Problems, Discrete Fourier Analysis, and Geometry

SDP Gaps for 2-to-1 and Other Label-Cover Variants

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