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A Variant of the VC-Dimension with Applications to Depth-3 Circuits

Authors Peter Frankl, Svyatoslav Gryaznov , Navid Talebanfard



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Author Details

Peter Frankl
  • Rényi Institute, Budapest, Hungary
Svyatoslav Gryaznov
  • Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic
  • St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, Russia
Navid Talebanfard
  • Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic

Acknowledgements

We are grateful to Vojtěch Rödl for discussions and to ITCS'22 reviewers for useful comments.

Cite AsGet BibTex

Peter Frankl, Svyatoslav Gryaznov, and Navid Talebanfard. A Variant of the VC-Dimension with Applications to Depth-3 Circuits. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 72:1-72:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITCS.2022.72

Abstract

We introduce the following variant of the VC-dimension. Given S ⊆ {0,1}ⁿ and a positive integer d, we define 𝕌_d(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 𝕌_d 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 Σ₃^k-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 Σ₃² 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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Mathematics of computing → Combinatoric problems
  • Mathematics of computing → Hypergraphs
Keywords
  • VC-dimension
  • Hypergraph
  • Clique
  • Affine Disperser
  • Circuit

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