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Some topics in analysis of boolean functions

Published: 17 May 2008 Publication History

Abstract

This article accompanies a tutorial talk given at the 40th ACM STOC conference. In it, we give a brief introduction to Fourier analysis of boolean functions and then discuss some applications: Arrow's Theorem and other ideas from the theory of Social Choice; the Bonami-Beckner Inequality as an extension of Chernoff/Hoeffding bounds to higher-degree polynomials; and, hardness for approximation algorithms.

References

[1]
G. Andersson and L. Engebretsen. Better approximation algorithms for Set Splitting Not-All-Equal-SAT. Inf. Proc. Lett., 65(6):305--311, 1998.
[2]
S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501--555, 1998.
[3]
S. Arora and S. Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70--122, 1998.
[4]
K. Arrow. A difficulty in the concept of social welfare. J. of Political Economy, 58(4):328--346, 1950.
[5]
J. Banzhaf. Weighted voting doesn't work: A mathematical analysis. Rutgers Law Review, 19(2):317--343, 1965.
[6]
W. Beckner. Inequalities in Fourier analysis. Ann. Math., 102(1):159--182, 1975.
[7]
M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCPs, and nonapproximability towards tight results. SICOMP, 27(3):804--915, 1998.
[8]
M. Ben-Or and N. Linial. Collective coin flipping, robust voting schemes and minima of Banzhaf values. In Proc. 26th FOCS, pages 408--416, 1985.
[9]
A. Bonami. ´Etude des coefficients de Fourier des fonctions de Lp(G). Ann. Inst. Fourier, 20(2):335--402, 1970.
[10]
N. de Condorcet. Essai sur l'application de l'analyse a la probabilite des d´ecisions rendues a la pluralite des voix. Imprimerie Royale, Paris, 1785.
[11]
D. Felsenthal and M. Machover. The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes. Edward Elgar, 1998.
[12]
E. Friedgut. Boolean functions with low average sensitivity depend on few coordinates. Combinatorica, 18(1):27--36, 1998.
[13]
E. Friedgut, G. Kalai, and A. Naor. Boolean functions whose Fourier transform is concentrated on the first two levels. Adv. in Appl. Math, 29(3):427--437, 2002.
[14]
M. Garman and M. Kamien. The paradox of voting: probability calculations. Behavioral Science, 13(4):306--16, 1968.
[15]
J. Geanakoplos. Three brief proofs of Arrow's Impossibility Theorem. Economic Theory, 26(1):211--215, 2005.
[16]
M. Goemans and D. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42:1115--1145, 1995.
[17]
L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math., 97:1061--1083, 1975.
[18]
G. Guilbaud. Les theories de l'interÆet general et le probleme logique de l'agregration. Economie appliquee, 5:501--584, 1952.
[19]
V. Guruswami. Inapproximability results for Set Splitting and satisfiability problems with no mixed clauses. Algorithmica, 38(3):451--469, 2003.
[20]
J. Hastad. Some optimal inapproximability results. J. ACM, 48(4):798--859, 2001.
[21]
S. Janson. Gaussian Hilbert Spaces, volume 129 of Cambridge Tracts in Mathematics. Cambridge University Press, 1997.
[22]
J. Kahn, G. Kalai, and N. Linial. The influence of variables on Boolean functions. In Proc. 29th FOCS, pages 68--80, 1988.
[23]
A. Kalai, A. Klivans, Y. Mansour, and R. Servedio. Agnostically learning halfspaces. In Proc. 46th FOCS, pages 11--20, 2005.
[24]
G. Kalai. A Fourier-theoretic perspective on the Concordet paradox and Arrow's theorem. Adv. in Appl. Math., 29(3):412--426, 2002.
[25]
G. Kalai. Noise sensitivity and chaos in social choice theory. Discussion Paper Series dp399, Center for Rationality and Interactive Decision Theory, Hebrew University, 2005.
[26]
V. Kann, J. Lagergren, and A. Panconesi. Approximability of maximum splitting of k-sets and some other APX-complete problems. Inf. Proc. Lett., 58(3):105--110, 1996.
[27]
M. Karpovsky. Finite orthogonal series in the design of digital devices. John Wiley, 1976.
[28]
S. Khot. On the power of unique 2-prover 1-round games. In Proc. 34th STOC, pages 767--775, 2002.
[29]
S. Khot. Inapproximability results via Long Code based PCPs. SIGACT News, 36(2):25--42, 2005.
[30]
S. Khot, G. Kindler, E. Mossel, and R. O'Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SICOMP, 37(1):319--357, 2007.
[31]
S. Khot and N. Vishnoi. The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into '1. In Proc. 46th FOCS, pages 53--62, 2005.
[32]
R. Krauthgamer and Y. Rabani. Improved lower bounds for embeddings into l1. In Proc. 17th SODA, pages 1010--1017, 2006.
[33]
Y. Mansour. An O(nlog log n) learning algorithm for DNF under the uniform distribution. J. Comput. Sys. Sci., 50(3):543--550, 1995.
[34]
E. Mossel, R. O'Donnell, and K. Oleszkiewicz. Noise stability of functions with low influences: invariance and optimality. In Proc. 46th FOCS, pages 21--30, 2005. To appear, Ann. Math.
[35]
L. Penrose. The elementary statistics of majority voting. J. of the Royal Statistical Society, 109(1):53--57, 1946.
[36]
Y. Peres. Noise stability of weighted majority. arXiv:math/0412377v1, 2004.
[37]
R. Raz. A parallel repetition theorem. SICOMP, 27(3):763--803, 1998.
[38]
W. Sheppard. On the application of the theory of error to cases of normal distribution and normal correlation. Phil. Trans. Royal Soc. London, Series A, 192:101--531, 1899.
[39]
M. Talagrand. On Russo's approximate zero-one law. Ann. Prob., 22(3):1576--1587, 1994.
[40]
L. Trevisan. Inapproximability of combinatorial optimization problems. Electronic Colloq. on Comp. Complexity (ECCC), 065, 2004.
[41]
U. Zwick. Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. In Proc. 9th SODA, pages 201--210, 1998.
[42]
U. Zwick. Outward rotations: A tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In Proc. 31st STOC, pages 679--687, 1999.

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      cover image ACM Conferences
      STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing
      May 2008
      712 pages
      ISBN:9781605580470
      DOI:10.1145/1374376
      Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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      Published: 17 May 2008

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      1. analysis of boolean functions
      2. fourier analysis

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      May 17 - 20, 2008
      British Columbia, Victoria, Canada

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