TY - GEN
T1 - Biasing boolean functions and collective coin-flipping protocols over arbitrary product distributions
AU - Filmus, Yuval
AU - Hambardzumyan, Lianna
AU - Hatami, Hamed
AU - Hatami, Pooya
AU - Zuckerman, David
N1 - Publisher Copyright:
© Graham Cormode, Jacques Dark, and Christian Konrad; licensed under Creative Commons License CC-BY
PY - 2019/7/1
Y1 - 2019/7/1
N2 - The seminal result of Kahn, Kalai and Linial shows that a coalition of O(lognn ) 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 to 1 − with respect to the uniform measure. This fails for general product distributions, as the example of the AND function with respect to µ1−1/n shows. Instead, we use a novel boosting argument alongside a generalization of our first result to arbitrary finite ranges.
AB - The seminal result of Kahn, Kalai and Linial shows that a coalition of O(lognn ) 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 to 1 − with respect to the uniform measure. This fails for general product distributions, as the example of the AND function with respect to µ1−1/n shows. Instead, we use a novel boosting argument alongside a generalization of our first result to arbitrary finite ranges.
KW - Boolean function analysis
KW - Coin flipping
UR - http://www.scopus.com/inward/record.url?scp=85069165433&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ICALP.2019.58
DO - 10.4230/LIPIcs.ICALP.2019.58
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AN - SCOPUS:85069165433
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019
A2 - Baier, Christel
A2 - Chatzigiannakis, Ioannis
A2 - Flocchini, Paola
A2 - Leonardi, Stefano
T2 - 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019
Y2 - 9 July 2019 through 12 July 2019
ER -