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Low-Sensitivity Functions from Unambiguous Certificates

Authors Shalev Ben-David, Pooya Hatami, Avishay Tal

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Shalev Ben-David
Pooya Hatami
Avishay Tal

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Shalev Ben-David, Pooya Hatami, and Avishay Tal. Low-Sensitivity Functions from Unambiguous Certificates. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 28:1-28:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ITCS.2017.28

Abstract

We provide new query complexity separations against sensitivity for total Boolean functions: a power 3 separation between deterministic (and even randomized or quantum) query complexity and sensitivity, and a power 2.22 separation between certificate complexity and sensitivity. We get these separations by using a new connection between sensitivity and a seemingly unrelated measure called one-sided unambiguous certificate complexity. We also show that one-sided unambiguous certificate complexity is lower-bounded by fractional block sensitivity, which means we cannot use these techniques to get a super-quadratic separation between block sensitivity and sensitivity.
    
Along the way, we give a power 1.22 separation between certificate complexity and one-sided unambiguous certificate complexity, improving the power 1.128 separation due to Goos [FOCS 2015]. As a consequence, we obtain an improved lower-bound on the co-nondeterministic communication complexity of the Clique vs. Independent Set problem.

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Keywords
  • Boolean functions
  • decision tree complexity
  • query complexity
  • sensitivity conjecture

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