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.
@InProceedings{bendavid_et_al:LIPIcs.ITCS.2017.28, author = {Ben-David, Shalev and Hatami, Pooya and Tal, Avishay}, title = {{Low-Sensitivity Functions from Unambiguous Certificates}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {28:1--28:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.28}, URN = {urn:nbn:de:0030-drops-81869}, doi = {10.4230/LIPIcs.ITCS.2017.28}, annote = {Keywords: Boolean functions, decision tree complexity, query complexity, sensitivity conjecture} }
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