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].
@InProceedings{manurangsi:LIPIcs.ITCS.2023.85, author = {Manurangsi, Pasin}, title = {{Improved Inapproximability of VC Dimension and Littlestone’s Dimension via (Unbalanced) Biclique}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {85:1--85:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.85}, URN = {urn:nbn:de:0030-drops-175884}, doi = {10.4230/LIPIcs.ITCS.2023.85}, annote = {Keywords: VC Dimension, Littlestone’s Dimension, Maximum Biclique, Hardness of Approximation, Fine-Grained Complexity} }
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