Improved Inapproximability of VC Dimension and Littlestone’s Dimension via (Unbalanced) Biclique

Author Pasin Manurangsi

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Pasin Manurangsi
  • Google Research, Bangkok, Thailand


I would like to thank ITCS 2023 reviewers for their helpful comments and suggestions.

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Pasin Manurangsi. Improved Inapproximability of VC Dimension and Littlestone’s Dimension via (Unbalanced) Biclique. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 85:1-85:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We study the complexity of computing (and approximating) VC Dimension and Littlestone’s Dimension when we are given the concept class explicitly. We give a simple reduction from Maximum (Unbalanced) Biclique problem to approximating VC Dimension and Littlestone’s Dimension. With this connection, we derive a range of hardness of approximation results and running time lower bounds. For example, under the (randomized) Gap-Exponential Time Hypothesis or the Strongish Planted Clique Hypothesis, we show a tight inapproximability result: both dimensions are hard to approximate to within a factor of o(log n) in polynomial-time. These improve upon constant-factor inapproximability results from [Pasin Manurangsi and Aviad Rubinstein, 2017].

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • VC Dimension
  • Littlestone’s Dimension
  • Maximum Biclique
  • Hardness of Approximation
  • Fine-Grained Complexity


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