Frankl, P., Gryaznov, S. and Talebanfard, N. (2022) A variant of the VC-dimension with applications to depth-3 circuits. In: Braverman, M., (ed.) 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), 31 Jan - 03 Feb 2022, Berkeley, CA, USA (Virtual). Leibniz International Proceedings in Informatics (LIPIcs), 215 . Schloss Dagstuhl - Leibniz-Zentrum für Informatik , 72:1-72:19. ISBN 9783959772174
Abstract
We introduce the following variant of the VC-dimension. Given S ⊆ {0,1}n and a positive integer d, we define Ud(S) to be the size of the largest subset I ⊆ [n] such that the projection of S on every subset of I of size d is the d-dimensional cube. We show that determining the largest cardinality of a set with a given Ud dimension is equivalent to a Turán-type problem related to the total number of cliques in a d-uniform hypergraph. This allows us to beat the Sauer-Shelah lemma for this notion of dimension. We use this to obtain several results on Σk3-circuits, i.e., depth-3 circuits with top gate OR and bottom fan-in at most k: Tight relationship between the number of satisfying assignments of a 2-CNF and the dimension of the largest projection accepted by it, thus improving Paturi, Saks, and Zane (Comput. Complex.'00). Improved Σ33-circuit lower bounds for affine dispersers for sublinear dimension. Moreover, we pose a purely hypergraph-theoretic conjecture under which we get further improvement. We make progress towards settling the Σ23 complexity of the inner product function and all degree-2 polynomials over F2 in general. The question of determining the Σ33 complexity of IP was recently posed by Golovnev, Kulikov, and Williams (ITCS'21).
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Editors: |
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Copyright, Publisher and Additional Information: | © 2022 Peter Frankl, Svyatoslav Gryaznov, and Navid Talebanfard. Licensed under Creative Commons License CC-BY 4.0. (https://creativecommons.org/licenses/by/4.0/) |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 18 Oct 2023 15:00 |
Last Modified: | 18 Oct 2023 15:32 |
Status: | Published |
Publisher: | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |
Series Name: | Leibniz International Proceedings in Informatics (LIPIcs) |
Refereed: | Yes |
Identification Number: | 10.4230/LIPIcs.ITCS.2022.72 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:204339 |