2026-07-21T14:14:25Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:10634 2024-03-06T10:46:20Z ddc:004 open_access

Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions Filmus, Yuval Hambardzumyan, Lianna Hatami, Hamed Hatami, Pooya Zuckerman, David Boolean function analysis coin flipping 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Yuval Filmus and Lianna Hambardzumyan and Hamed Hatami and Pooya Hatami and David Zuckerman 2019 Is Part Of LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2019.58 urn:nbn:de:0030-drops-106340 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.58 eng https://creativecommons.org/licenses/by/3.0/legalcode