research-article

On the Complexity of Random Satisfiability Problems with Planted Solutions

Authors:

Vitaly Feldman,

Santosh VempalaAuthors Info & Claims

STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of Computing

Pages 77 - 86

https://doi.org/10.1145/2746539.2746577

Published: 14 June 2015 Publication History

Abstract

The problem of identifying a planted assignment given a random k-SAT formula consistent with the assignment exhibits a large algorithmic gap: while the planted solution can always be identified given a formula with O(n log n) clauses, there are distributions over clauses for which the best known efficient algorithms require n^k/2 clauses. We propose and study a unified model for planted k-SAT, which captures well-known special cases. An instance is described by a planted assignment σ and a distribution on clauses with k literals. We define its distribution complexity as the largest r for which the distribution is not r-wise independent (1 ≤ r ≤ k for any distribution with a planted assignment).

Our main result is an unconditional lower bound, tight up to logarithmic factors, of Ω(n^r/2) clauses for statistical algorithms, matching the known upper bound (which, as we show, can be implemented using a statistical algorithm). Since known approaches for problems over distributions have statistical analogues (spectral, MCMC, gradient-based, convex optimization etc.), this lower bound provides a rigorous explanation of the observed algorithmic gap. The proof introduces a new general technique for the analysis of statistical algorithms. It also points to a geometric paring phenomenon in the space of all planted assignments.

We describe consequences of our lower bounds to Feige's refutation hypothesis and to lower bounds on general convex programs that solve planted k-SAT. Our bounds also extend to the planted k-CSP model, defined by Goldreich as a candidate for one-way function, and therefore provide concrete evidence for the security of Goldreich's one-way function and the associated pseudorandom generator when used with a sufficiently hard predicate.

References

[1]

D. Achlioptas and A. Coja-Oghlan. Algorithmic barriers from phase transitions. In Foundations of Computer Science, 2008. FOCS'08. IEEE 49th Annual IEEE Symposium on, pages 793--802. IEEE, 2008.

Digital Library

[2]

D. Achlioptas, H. Jia, and C. Moore. Hiding satisfying assignments: Two are better than one. J. Artif. Intell. Res.(JAIR), 24:623--639, 2005.

Digital Library

[3]

M. Alekhnovich. More on average case vs approximation complexity. Computational Complexity, 20(4):755--786, 2011.

Digital Library

[4]

B. Applebaum. Pseudorandom generators with long stretch and low locality from random local one-way functions. In Proceedings of the 44th symposium on Theory of Computing, pages 805--816. ACM, 2012.

Digital Library

[5]

B. Applebaum, B. Barak, and A. Wigderson. Public-key cryptography from different assumptions. In Proceedings of the 42nd ACM symposium on Theory of computing, pages 171--180. ACM, 2010.

Digital Library

[6]

B. Applebaum, A. Bogdanov, and A. Rosen. A dichotomy for local small-bias generators. In Theory of Cryptography, pages 600--617. Springer, 2012.

Digital Library

[7]

B. Barak, G. Kindler, and D. Steurer. On the optimality of semidefinite relaxations for average-case and generalized constraint satisfaction. In Proceedings of the 4th conference on Innovations in Theoretical Computer Science, pages 197--214. ACM, 2013.

Digital Library

[8]

W. Barthel, A. K. Hartmann, M. Leone, F. Ricci-Tersenghi, M. Weigt, and R. Zecchina. Hiding solutions in random satisfiability problems: A statistical mechanics approach. Physical review letters, 88(18):188701, 2002.

[9]

W. Beckner. Inequalities in fourier analysis. The Annals of Mathematics, 102(1):159--182, 1975.

[10]

A. Belloni, R. M. Freund, and S. Vempala. An efficient rescaled perceptron algorithm for conic systems. Math. Oper. Res., 34(3):621--641, 2009.

Digital Library

[11]

J. Blocki, M. Blum, A. Datta, and S. Vempala. Human computable passwords. CoRR, abs/1404.0024, 2014.

[12]

A. Blum, C. Dwork, F. McSherry, and K. Nissim. Practical privacy: the SuLQ framework. In Proceedings of PODS, pages 128--138, 2005.

Digital Library

[13]

A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, and S. Rudich. Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proceedings of STOC, pages 253--262, 1994.

Digital Library

[14]

A. Bogdanov and Y. Qiao. On the security of goldreich's one-way function. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 392--405. Springer, 2009.

Digital Library

[15]

A. Bonami. Étude des coefficients de fourier des fonctions de l_p(g). In Annales de l'institut Fourier, volume 20, pages 335--402. Institut Fourier, 1970.

[16]

R. B. Boppana. Eigenvalues and graph bisection: An average-case analysis. In Foundations of Computer Science, 1987., 28th Annual Symposium on, pages 280--285. IEEE, 1987.

Digital Library

[17]

M. Charikar and A. Wirth. Maximizing quadratic programs: Extending grothendieck's inequality. In FOCS, pages 54--60, 2004.

Digital Library

[18]

C.-T. Chu, S. K. Kim, Y.-A. Lin, Y. Yu, G. Bradski, A. Y. Ng, and K. Olukotun. Map-reduce for machine learning on multicore. In Proceedings of NIPS, pages 281--288, 2006.

[19]

A. Coja-Oghlan. A spectral heuristic for bisecting random graphs. Random Structures & Algorithms, 29:3:351--398, 2006.

Digital Library

[20]

A. Coja-Oghlan, C. Cooper, and A. Frieze. An efficient sparse regularity concept. SIAM Journal on Discrete Mathematics, 23(4):2000--2034, 2010.

Digital Library

[21]

A. Coja-Oghlan, A. Goerdt, A. Lanka, and F. Sch\"adlich. Techniques from combinatorial approximation algorithms yield efficient algorithms for random 2k-sat. Theoretical Computer Science, 329(1):1--45, 2004.

Digital Library

[22]

J. Cook, O. Etesami, R. Miller, and L. Trevisan. Goldreich's one-way function candidate and myopic backtracking algorithms. In Theory of Cryptography, pages 521--538. Springer, 2009.

Digital Library

[23]

A. Dempster, N. Laird, and D. Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society, Series B, 39(1):1--38, 1977.

[24]

I. Dinur, E. Friedgut, G. Kindler, and R. O'Donnell. On the fourier tails of bounded functions over the discrete cube. Israel Journal of Mathematics, 160(1):389--412, 2007.

[25]

J. Dunagan and S. Vempala. A simple polynomial-time rescaling algorithm for solving linear programs. Math. Program., 114(1):101--114, 2008.

Digital Library

[26]

U. Feige. Relations between average case complexity and approximation complexity. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 534--543. ACM, 2002.

Digital Library

[27]

U. Feige and E. Ofek. Easily refutable subformulas of large random 3cnf formulas. In Automata, languages and programming, pages 519--530. Springer, 2004.

[28]

V. Feldman. Unpublished manuscript.

[29]

V. Feldman. A complete characterization of statistical query learning with applications to evolvability. Journal of Computer System Sciences, 78(5):1444--1459, 2012.

Digital Library

[30]

V. Feldman, E. Grigorescu, L. Reyzin, S. Vempala, and Y. Xiao. Statistical algorithms and a lower bound for planted clique. In Proceedings of the 45th annual ACM symposium on Symposium on theory of computing, pages 655--664. ACM, 2013.

Digital Library

[31]

V. Feldman, W. Perkins, and S. Vempala. Subsampled power iteration: a unified algorithm for block models and planted csp's. arXiv preprint arXiv:1407.2774, 2014.

[32]

A. Flaxman. A spectral technique for random satisfiable 3cnf formulas. In Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, pages 357--363. Society for Industrial and Applied Mathematics, 2003.

Digital Library

[33]

J. Friedman, A. Goerdt, and M. Krivelevich. Recognizing more unsatisfiable random k-sat instances efficiently. SIAM Journal on Computing, 35(2):408--430, 2005.

Digital Library

[34]

D. Gamarnik and M. Sudan. Performance of the survey propagation-guided decimation algorithm for the random nae-k-sat problem. arXiv preprint arXiv:1402.0052, 2014.

[35]

A. E. Gelfand and A. F. Smith. Sampling based approaches to calculating marginal densities. Journal of the American Statistical Association, 85:398--409, 1990.

[36]

A. Goerdt and A. Lanka. Recognizing more random unsatisfiable 3-sat instances efficiently. Electronic Notes in Discrete Mathematics, 16:21--46, 2003.

[37]

O. Goldreich. Candidate one-way functions based on expander graphs. IACR Cryptology ePrint Archive, 2000:63, 2000.

[38]

L. P. Hansen. Large sample properties of generalized method of moments estimators. Econometrica, 50:1029--1054, 2012.

[39]

Y. Ishai, E. Kushilevitz, R. Ostrovsky, and A. Sahai. Cryptography with constant computational overhead. In Proceedings of the 40th annual ACM symposium on Theory of computing, pages 433--442. ACM, 2008.

Digital Library

[40]

S. Janson. Gaussian Hilbert spaces. Cambridge University Press, 1997.

[41]

H. Jia, C. Moore, and D. Strain. Generating hard satisfiable formulas by hiding solutions deceptively. In PROCEEDINGS OF THE NATIONAL CONFERENCE ON ARTIFICIAL INTELLIGENCE, volume 20, page 384. Menlo Park, CA; Cambridge, MA; London; AAAI Press; MIT Press; 1999, 2005.

Digital Library

[42]

M. Kearns. Efficient noise-tolerant learning from statistical queries. Journal of the ACM (JACM), 45(6):983--1006, 1998.

Digital Library

[43]

S. Kirkpatrick, D. G. Jr., and M. P. Vecchi. Optimization by simmulated annealing. Science, 220(4598):671--680, 1983.

[44]

F. Krzakala, M. Mézard, and L. Zdeborová. Reweighted belief propagation and quiet planting for random k-sat. arXiv preprint arXiv:1203.5521, 2012.

[45]

F. Krzakała, A. Montanari, F. Ricci-Tersenghi, G. Semerjian, and L. Zdeborová. Gibbs states and the set of solutions of random constraint satisfaction problems. Proceedings of the National Academy of Sciences, 104(25):10318--10323, 2007.

[46]

F. Krzakala and L. Zdeborová. Hiding quiet solutions in random constraint satisfaction problems. Physical review letters, 102(23):238701, 2009.

[47]

F. McSherry. Spectral partitioning of random graphs. In Foundations of Computer Science, 2001. Proceedings. 42nd IEEE Symposium on, pages 529--537. IEEE, 2001.

Digital Library

[48]

E. Mossel, R. O'Donnell, and R. Servedio. Learning functions of k relevant variables. Journal of Computer and System Sciences, 69(3):421--434, 2004.

Digital Library

[49]

E. Mossel, A. Shpilka, and L. Trevisan. On ε-biased generators in nc0. Random Structures & Algorithms, 29(1):56--81, 2006.

Digital Library

[50]

R. O'Donnell and D. Witmer. Goldreich's prg: Evidence for near-optimal polynomial stretch. In Conference on Computational Complexity, 2014.

Digital Library

[51]

M. A. Tanner and W. H. Wong. The calculation of posterior distributions by data augmentation (with discussion). Journal of the American Statistical Association, 82:528--550, 1987.

[52]

L. Trevisan. Checking the quasirandomness of graphs and hypergraphs. http://terrytao.wordpress.com/2008/02/15/luca-trevisan-checking-the-quasirandomness-of-graphs-and-hypergraphs/, February 2008.

[53]

V. Cerný. Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. Journal of Optimization Theory and Applications, 45(1):41--51, Jan. 1985.

Digital Library

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Index Terms

On the Complexity of Random Satisfiability Problems with Planted Solutions
1. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of Computing

June 2015

916 pages

ISBN:9781450335362

DOI:10.1145/2746539

General Chair:
Rocco Servedio
Columbia University
,
Program Chair:
Ronitt Rubinfeld
MIT and Tel Aviv University

Copyright © 2015 ACM.

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Published: 14 June 2015

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Bresler GGuo CPolyanskiy Y(2023)Algorithmic Decorrelation and Planted Clique in Dependent Random Graphs: The Case of Extra Triangles2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00132(2149-2158)Online publication date: 6-Nov-2023
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