# Inapproximability of VC Dimension and Littlestone's Dimension

## TL;DR

This paper investigates the computational complexity of determining VC and Littlestone's Dimensions, establishing quasi-polynomial algorithms and ETH-based lower bounds that highlight their inapproximability.

## Contribution

It provides the first quasi-polynomial time algorithms and matching ETH-based lower bounds for approximating VC and Littlestone's Dimensions.

## Key findings

- Exact computation is in quasi-polynomial time.
- Under ETH, approximation is nearly as hard as exact computation.
- The results establish fundamental inapproximability bounds.

## 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 approximation algorithms.