Document

# Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions

## File

LIPIcs.ICALP.2019.58.pdf
• Filesize: 478 kB
• 13 pages

## Acknowledgements

Part of the work on this paper was done while the first three authors were at the Simons Institute for the Theory of Computing at Berkeley, CA, USA.

## Cite As

Yuval Filmus, Lianna Hambardzumyan, Hamed Hatami, Pooya Hatami, and David Zuckerman. Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 58:1-58:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.58

## Abstract

The seminal result of Kahn, Kalai and Linial shows that a coalition of O(n/(log n)) players can bias the outcome of any Boolean function {0,1}^n -> {0,1} with respect to the uniform measure. We extend their result to arbitrary product measures on {0,1}^n, by combining their argument with a completely different argument that handles very biased input bits. We view this result as a step towards proving a conjecture of Friedgut, which states that Boolean functions on the continuous cube [0,1]^n (or, equivalently, on {1,...,n}^n) can be biased using coalitions of o(n) players. This is the first step taken in this direction since Friedgut proposed the conjecture in 2004. Russell, Saks and Zuckerman extended the result of Kahn, Kalai and Linial to multi-round protocols, showing that when the number of rounds is o(log^* n), a coalition of o(n) players can bias the outcome with respect to the uniform measure. We extend this result as well to arbitrary product measures on {0,1}^n. The argument of Russell et al. relies on the fact that a coalition of o(n) players can boost the expectation of any Boolean function from epsilon to 1-epsilon with respect to the uniform measure. This fails for general product distributions, as the example of the AND function with respect to mu_{1-1/n} shows. Instead, we use a novel boosting argument alongside a generalization of our first result to arbitrary finite ranges.

## Subject Classification

##### ACM Subject Classification
• Theory of computation
##### Keywords
• Boolean function analysis
• coin flipping

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Miklós Ajtai and Nathan Linial. The influence of large coalitions. Combinatorica, 13(2):129-145, 1993.
2. Michael Ben-Or and Nathan Linial. Collective coin flipping. Advances in Computing Research, 5:91-115, 1989.
3. Michael Ben-Or, Nathan Linial, and Michael Saks. Collective coin flipping and other models of imperfect randomness. IBM Thomas J. Watson Research Division, 1989.
4. Jean Bourgain, Jeff Kahn, Gil Kalai, Yitzhak Katznelson, and Nathan Linial. The influence of variables in product spaces. Israel Journal of Mathematics, 77(1-2):55-64, 1992.
5. Eshan Chattopadhyay and David Zuckerman. Explicit two-source extractors and resilient functions. Annals of Mathematics, to appear, 2016. Preliminary version in STOC 2016.
6. Benny Chor and Cynthia Dwork. Randomization in Byzantine Agreement. Advances in Computing Research, 5:443-497, 1989.
7. Yevgeniy Dodis. Fault-tolerant leader election and collective coin-flipping in the full information model. Survey, 2006.
8. Uriel Feige. Noncryptographic Selection Protocols. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science, page 142. IEEE Computer Society, 1999.
9. Ehud Friedgut. Influences in Product Spaces: KKL and BKKKL Revisited. Combinatorics, Probability and Computing, 13(1):17-29, 2004.
10. Jeff Kahn, Gil Kalai, and Nathan Linial. The influence of variables on Boolean functions. In Proceedings of the 29th annual FOCS, pages 68-80, 1988.
11. Raghu Meka. Explicit resilient functions matching Ajtai-Linial. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1132-1148. SIAM, 2017.
12. Alexander Russell, Michael Saks, and David Zuckerman. Lower bounds for leader election and collective coin-flipping in the perfect information model. SIAM Journal on Computing, 31(6):1645-1662, 2002.
13. Alexander Russell and David Zuckerman. Perfect information leader election in log* n+ O (1) rounds. Journal of Computer and System Sciences, 63(4):612-626, 2001.
14. Terence Tao. Soft analysis, hard analysis, and the finite convergence principle. URL: https://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/, 2007. Accessed 10 Feb 2019.
X

Feedback for Dagstuhl Publishing