KKL, Kruskal-Katona, and Monotone Nets

We generalize the Kahn-Kalai-Linial (KKL) Theorem to random walks on Cayley and Schreier graphs, making progress on an open problem of Hoory, Linial, and Wigderson. In our generalization, the underlying group need not be abelian so long as the generating set is a union of conjugacy classes.

[1]  Terence Tao,et al.  Structure and Randomness in Combinatorics , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[2]  John Langford,et al.  On learning monotone Boolean functions , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[3]  G. Grimmett,et al.  Influence and sharp-threshold theorems for monotonic measures , 2005, math/0505057.

[4]  E. Friedgut,et al.  Sharp thresholds of graph properties, and the -sat problem , 1999 .

[5]  Nathan Linial,et al.  The influence of variables on Boolean functions , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[6]  P. Diaconis,et al.  Generating a random permutation with random transpositions , 1981 .

[7]  L. Gross.  LOGARITHMIC SOBOLEV INEQUALITIES. , 1975 .

[8]  Y. Rabani,et al.  Improved lower bounds for embeddings into L 1 , 2006, SODA 2006.

[9]  L. Russo.  An approximate zero-one law , 1982 .

[10]  Yuval Rabani,et al.  On the Hardness of Approximating Multicut and Sparsest-Cut , 2005, Computational Complexity Conference.

[11]  David Zuckerman,et al.  DETERMINISTIC EXTRACTORS FOR BIT-FIXING SOURCES AND EXPOSURE-RESILIENT CRYPTOGRAPHY , 2003 .

[12]  Yuval Rabani,et al.  Improved lower bounds for embeddings into L1 , 2006, SODA '06.

[13]  Akira Maruoka,et al.  On Learning Monotone Boolean Functions under the Uniform Distribution , 2002, ALT.

[14]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[15]  Guan-Yu Chen,et al.  On the log-Sobolev constant for the simple random walk on the n-cycle: the even cases , 2003 .

[16]  Ehud Friedgut,et al.  Boolean Functions With Low Average Sensitivity Depend On Few Coordinates , 1998, Comb..

[17]  Leslie G. Valiant,et al.  Cryptographic limitations on learning Boolean formulae and finite automata , 1994, JACM.

[18]  Peter Keevash,et al.  Shadows and intersections: Stability and new proofs , 2008, 0806.2023.

[19]  M. Talagrand.  On Russo's Approximate Zero-One Law , 1994 .

[20]  Spyridon Antonakopoulos,et al.  Buy-at-Bulk Network Design with Protection , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[21]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[22]  S. Safra,et al.  On the hardness of approximating minimum vertex cover , 2005 .

[23]  P. Diaconis,et al.  Time to reach stationarity in the Bernoulli-Laplace diffusion model , 1987 .

[24]  P. Diaconis,et al.  LOGARITHMIC SOBOLEV INEQUALITIES FOR FINITE MARKOV CHAINS , 1996 .

[25]  N. Linial,et al.  The influence of variables in product spaces , 1992 .

[26]  Ran Raz,et al.  Fourier analysis for probabilistic communication complexity , 1995, computational complexity.

[27]  Nathan Linial,et al.  Collective Coin Flipping , 1989, Adv. Comput. Res..

[28]  Raphael Rossignol.  Threshold for monotone symmetric properties through a logarithmic Sobolev inequality , 2005, math/0511607.

[29]  I. Benjamini,et al.  Noise sensitivity of Boolean functions and applications to percolation , 1998 .

[30]  Dvir Falik,et al.  Edge-Isoperimetric Inequalities and Influences , 2007, Comb. Probab. Comput..

[31]  G. Katona.  A theorem of finite sets , 2009 .

[32]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[33]  Tzong-Yow Lee,et al.  Logarithmic Sobolev inequality for some models of random walks , 1998 .

[34]  Nisheeth K. Vishnoi,et al.  Integrality gaps for sparsest cut and minimum linear arrangement problems , 2006, STOC '06.

[35]  D. Kleitman.  Families of Non-disjoint subsets* , 1966 .

[36]  Béla Bollobás,et al.  Threshold functions , 1987, Comb..

[37]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[38]  Nader H. Bshouty,et al.  On the Fourier spectrum of monotone functions , 1995, STOC '95.

[39]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[40]  Ryan O'Donnell,et al.  Learning Monotone Decision Trees in Polynomial Time , 2007, SIAM J. Comput..

[41]  G. Kalai,et al.  Every monotone graph property has a sharp threshold , 1996 .

[42]  Jonathan L. Gross.  Every connected regular graph of even degree is a Schreier coset graph , 1977, J. Comb. Theory, Ser. B.