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

TL;DR

This paper establishes stronger hardness of approximation results for VC and Littlestone's Dimensions by reducing from the (Unbalanced) Biclique problem, showing they are hard to approximate within a factor of o(log n) under certain hypotheses.

Contribution

It introduces a simple reduction from the (Unbalanced) Biclique problem to the approximation of VC and Littlestone's Dimensions, leading to improved inapproximability bounds.

Findings

Hardness of approximation within o(log n) under Gap-Exponential Time Hypothesis

Polynomial-time inapproximability results surpass previous constant-factor bounds

Establishes tight lower bounds based on the Strongish Planted Clique Hypothesis

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].