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

• URL: http://arxiv.org/abs/2211.01443v1
• Date: Wed, 2 Nov 2022 19:23:42 GMT
• Title: Improved Inapproximability of VC Dimension and Littlestone's Dimension via (Unbalanced) Biclique
• Authors: Pasin Manurangsi
• Abstract summary: 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.
• Score: 28.57552551316786
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 [Manurangsi and Rubinstein, COLT 2017].

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