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# Inapproximability of VC Dimension and Littlestone’s Dimension

*Proceedings of the 2017 Conference on Learning Theory*, PMLR 65:1432-1460, 2017.

#### Abstract

We study the complexity of computing the VC Dimension and Littlestone’s Dimension. Given an explicit description of a finite universe and a concept class (a binary matrix whose $(x,C)$-th entry is $1$ iff element $x$ belongs to concept $C$), both can be computed exactly in quasi-polynomial time ($n^O(\log n)$). Assuming the randomized Exponential Time Hypothesis (ETH), we prove nearly matching lower bounds on the running time, that hold even for \em approximation algorithms.