Linear Level Lasserre Lower Bounds for Certain k-CSPs

Schoenebeck, Grant

doi:10.1109/focs.2008.74

Cited by 215 publications

(232 citation statements)

References 18 publications

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“…Proof: Note that Schoenebeck [37] has proven precisely this theorem in the case that k = t; i.e., when P is simply the t-ary XOR predicate. 1 In particular, one can view his proof as constructing an appropriate "pseudoexpectation" operator E[·] (see [22]) for degree2r polynomials under which all t-XOR constraints are satisfied with "pseudoprobability 1".…”

Section: Security Against Lasserre/parrilo Attacksmentioning

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: Security Against Lasserre/parrilo Attacksmentioning

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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“…This situation is similar to the very strong SDP gaps known for problems such as 3-XOR (see [14], [17]) for which deciding complete satisfiability is easy.…”

Section: Our Resultsmentioning

confidence: 70%

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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“…Random CSP is extensively studied due to its importance to average-case complexity, probability and statistical physics [MZK + 99], cryptography [ABW10], hardness of approximation [Fei02] and other fields. It is also a prototypical example of a hard CSP [Gri01,Sch08]. Random Label-Cover, i.e., random approximate CSP, is therefore a natural object for study.…”

Section: Label-covermentioning

confidence: 99%

Approximation Algorithms for Label Cover and The Log-Density Threshold

Chlamtáč¹,

Manurangsi²,

Moshkovitz³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Many known optimal NP-hardness of approximation results are reductions from a problem called LabelCover. The input is a bipartite graph G = (L, R, E) and each edge e = (x, y) ∈ E carries a projection π e that maps labels to x to labels to y. The objective is to find a labeling of the vertices that satisfies as many of the projections as possible. It is believed that the best approximation ratio efficiently achievable for Label-Cover is of the form N −c where N = nk, n is the number of vertices, k is the number of labels, and 0 < c < 1 is some constant.Inspired by a framework originally developed for Densest k-Subgraph, we propose a "log density threshold" for the approximability of Label-Cover. Specifically, we suggest the possibility that the Label-Cover approximation problem undergoes a computational phase transition at the same threshold at which local algorithms for its random counterpart fail. This threshold is N 3−2 √ 2 ≈ N −0.17 . We then design, for any ε > 0, a polynomial-time approximation algorithm for semirandom Label-Cover whose approximation ratio is N 3−2 √ 2+ε . In our semi-random model, the input graph is random (or even just expanding), and the projections on the edges are arbitrary. time algorithm whose approximation ratio is roughly N −0.233 . The previous best efficient approximation ratio was N −0.25 . We present some evidence towards an N −c threshold by constructing integrality gaps for N Ω(1) rounds of the Sum-of-squares/Lasserre hierarchy of the natural relaxation of Label Cover. For general 2CSP the "log density threshold" is N −0.25 , and we give a polynomial-time algorithm in the semi-random model whose approximation ratio is N −0.25+ε for any ε > 0.

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Linear Level Lasserre Lower Bounds for Certain k-CSPs

Cited by 215 publications

References 18 publications

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

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

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

Approximation Algorithms for Label Cover and The Log-Density Threshold

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