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Maximally stable Gaussian partitions with discrete applications

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Abstract

Gaussian noise stability results have recently played an important role in proving results in hardness of approximation in computer science and in the study of voting schemes in social choice. We prove a new Gaussian noise stability result generalizing an isoperimetric result by Borell on the heat kernel and derive as applications:

  • An optimality result for majority in the context of Condorcet voting.

  • A proof of a conjecture on “cosmic coin tossing” for low influence functions.

We also discuss a Gaussian noise stability conjecture which may be viewed as a generalization of the “Double Bubble” theorem and show that it implies:

  • A proof of the “Plurality is Stablest Conjecture”.

  • That the Frieze-Jerrum SDP for MAX-q-CUT achieves the optimal approximation factor assuming the Unique Games Conjecture.

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References

  1. K. Arrow, A difficulty in the theory of social welfare, Journal of Political Economy 58 (1950), 328–346.

    Article  Google Scholar 

  2. P. Austrin, Balanced Max 2-Sat might not be the hardest, in Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, 2007, pp. 189–197.

  3. P. Austrin, Towards sharp inapproximability for any 2-CSP, in 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), Providence, RI, USA, Proceedings, 2007, pp. 307–317.

  4. C. Borell, Geometric bounds on the Ornstein-Uhlenbeck velocity process, Probability Theory and Related Fields 70 (1985), 1–13.

    MathSciNet  MATH  Google Scholar 

  5. A. Burchard and M. Schmuckenschlager, Comparison theorems for exit times, Geometric and Functional Analysis 11 (2001), 651–692.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Chawla, R. Krauthgamer, R. Kumar, Y. Rabani and D. Sivakumar, On the hardness of approximating multicut and sparsest-cut, in CCC’ 05: Proceedings of the 20th Annual IEEE Conference on Computational Complexity, IEEE Computer Society, Washington, DC, 2005, pp. 144–153

    Chapter  Google Scholar 

  7. J. Corneli, I. Corwin, Y. Xu, S. Hurder, V. Sesum, E. Adams, D. Davis, M. Lee, R. Pettit and N. Hoffman, Double bubbles in gauss space and spheres, Houston Journal of Mathematics 34 (2008), 181–204.

    MathSciNet  MATH  Google Scholar 

  8. E. de Klerk, D. V. Pasechnik and J. P. Warners, On approximate graph colouring and max-k-cut algorithms based on the theta-function, Journal of Combinatorial Optimization 8 (2004), 267–294.

    Article  MathSciNet  MATH  Google Scholar 

  9. I. Dinur, E. Mossel and O. Regev, Conditional hardness for approximate coloring, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, 2006, pp. 344–353.

  10. R. Dobrushin, P. Groeneboom and M. Ledoux, Lectures on Probability Theory and Statistics, Volume 71648 of Lecture Notes in Mathematics, Lectures from the 24th Saint-Flour Summer School held July 7–23, 1994, (P. Bernard, eds.), Springer-Verlag, Berlin, 1996.

    Google Scholar 

  11. R. Durrett, Probability: Theory and Examples, Duxbury Advanced Series, 2005.

  12. A. Frieze and M. Jerrum, Improved approximation algorithms for MAX-k-CUT and MAX-BISECTION, in Integer Programming and Combinatorial Optimization, Volume 920, (Egon Balas and Jens Clausen, eds.), Springer, Berlin, 1995, pp. 1–13.

    Chapter  Google Scholar 

  13. M. Goemans and D. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, Journal of the Association for Computing Machinery 42 (1995), 1115–1145.

    Article  MathSciNet  MATH  Google Scholar 

  14. V. Guruswami, R. Manokaran and P. Raghavendra, Beating the random ordering is hard: Inapproximability of maximum acyclic subgraph, in FOCS’ 08: Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Washington, DC, 2008, pp. 573–582.

    Chapter  Google Scholar 

  15. J. Håstad, Some optimal inapproximability results, in Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC), May 1997, pp. 1–10.

  16. M. Hutchings, F. Morgan, M. Ritor and A. Ros, Proof of the double bubble conjecture, Annals of Mathematics 155 (2002), 459–489.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Janson, Gaussian Hilbert Spaces, Volume 129 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1997.

    Book  MATH  Google Scholar 

  18. G. Kalai, A Fourier-theoretic perspective on the Concordet paradox and Arrow’s theorem, Advances in Applied Mathematics 29 (2002), 412–426.

    Article  MathSciNet  MATH  Google Scholar 

  19. G. Kalai, Social indeterminacy, Econometrica 72 (2004), 1565–1581.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Khot, On the power of unique 2-prover 1-round games, in STOC’ 02: Proceedings of the Thirty-fourth Annual ACM, Symposium on Theory of Computing, ACM New York, 2002, pp. 767–775.

  21. S. Khot, G. Kindler, E. Mossel and R. O’Donnell, Optimal inapproximability results for max-cut and other 2-variable csps? in 45th Symposium on Foundations of Computer Science (FOCS 2004), Rome, Italy, Proceedings, 2004, pp. 146–154.

  22. S. Khot, G. Kindler, E. Mossel and R. O’Donnell, Optimal inapproximability results for max-cut and other 2-variable csps? SIAM Journal on Computing 37 (2007), 319–357.

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Khot and O. Regev, Vertex cover might be hard to approximate to within 2−ɛ, in 18th Annual IEEE Conference on Computational Complexity (Complexity 2003), Aarhus, Denmark, 2003, pp. 379–388.

  24. S. Khot and N. Vishnoi, The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into l1, in 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), Pittsburgh, PA, USA, 2005, pp. 53–62.

  25. C. Morpurgo, Sharp inequalities for functional integrals and traces of conformally invariant operators, Duke Mathematical Journal 114 (2002), 477–453.

    Article  MathSciNet  MATH  Google Scholar 

  26. E. Mossel, Gaussian bounds for noise correlation of functions, Geometric and Functional Analysis 19 (2010), 1713–1756.

    Article  MathSciNet  MATH  Google Scholar 

  27. E. Mossel and R. O’Donnell, Coin flipping from a cosmic source: On error correction of truly random bits, Random Structures Algorithms 26 (2005), 418–436.

    Article  MathSciNet  MATH  Google Scholar 

  28. E. Mossel, R. O’Donnell and K. Oleszkiewicz, Noise stability of functions with low influences: invariance and optimality (extended abstract), in 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), 23–25 October 2005, Pittsburgh, PA, USA, Proceedings, IEEE Computer Society, 2005, pp. 21–30.

  29. E. Mossel, R. O’Donnell and K. Oleszkiewicz, Noise stability of functions with low influences: invariance and optimality, Annals of Mathematics 171 (2010), 295–341.

    Article  MathSciNet  MATH  Google Scholar 

  30. E. Mossel, R. O’Donnell, O. Regev, J. E. Steif and B. Sudakov, Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality, Israel Journal of Mathematics 154 (2006), 299–336.

    Article  MathSciNet  MATH  Google Scholar 

  31. R. O’Donnell and Y. Wu, An optimal SDP algorithm for Max-Cut, and equally optimal Long Code tests, in Proceedings of the 40th Annual Symposium on Theory of Computing (STOC) Victoria, British Columbia, Canada, 2008, pp. 335–344.

  32. P. Raghavendra, Optimal algorithms and inapproximability results for every CSP? in Proceedings of the 40th Annual Symposium on Theory of Computing (STOC) Victoria, British Columbia, Canada, 2008, pp. 245–254.

  33. V. I. Rotar’, Limit theorems for multilinear forms and quasipolynomial functions, Theory of Probability and its Applications 20 (1975), 512–532.

    Google Scholar 

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Authors and Affiliations

  1. Chalmers University of Technology and Göteborg University, SE-41296, Göteborg, Sweden

    Marcus Isaksson

  2. Department of Statistics, University of California, 423 Evans Hall, Berkeley, CA, 94720-3860, USA

    Elchanan Mossel

  3. Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Elchanan Mossel

Authors
  1. Marcus Isaksson
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  2. Elchanan Mossel
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Correspondence to Marcus Isaksson.

Additional information

Supported by an Alfred Sloan fellowship in Mathematics, by NSF CAREER grant DMS-0548249 (CAREER), by DOD ONR grant (N0014-07-1-05-06), by BSF grant 2004105, by ISF grant 1300/08, by a Minerva Foundation grant and by ERC Marie Curie Grant 2008 239317.

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Isaksson, M., Mossel, E. Maximally stable Gaussian partitions with discrete applications. Isr. J. Math. 189, 347–396 (2012). https://doi.org/10.1007/s11856-011-0181-7

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  • DOI: https://doi.org/10.1007/s11856-011-0181-7

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