On the Converse of Talagrand's Influence Inequality

In 1994, Talagrand showed a generalization of the celebrated KKL theorem. In this work, we prove that the converse of this generalization also holds. Namely, for any sequence of numbers $0 0$, it is possible to find a roughly balanced Boolean function $f$ such that $\textrm{Inf}_j[f] < a_j$ for every $1 \le j \le n$.

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

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

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

[4]  Ran Raz,et al.  A parallel repetition theorem , 1995, STOC '95.

[5]  Ryan O'Donnell,et al.  KKL, Kruskal-Katona, and Monotone Nets , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

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

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

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

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

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

[11]  Yuval Rabani,et al.  ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

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

[13]  Ryan O'Donnell,et al.  The chow parameters problem , 2008, SIAM J. Comput..

[14]  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..

[15]  Ehud Friedgut,et al.  Influences in Product Spaces: KKL and BKKKL Revisited , 2004, Combinatorics, Probability and Computing.

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