Parameterized and Approximation Complexity of Partial VC Dimension

TL;DR

This paper studies the computational complexity of Partial VC Dimension, a problem generalizing VC Dimension and Test Cover, including new variants and analyzing their difficulty on various hypergraph classes.

Contribution

It introduces Partial VC Dimension and Max Partial VC Dimension, exploring their complexity and providing insights into their difficulty on different hypergraph structures.

Findings

Partial VC Dimension generalizes VC Dimension and Test Cover.

Complexity results vary between general and restricted hypergraph classes.

New variants and problem formulations are proposed and analyzed.

Abstract

We introduce the problem Partial VC Dimension that asks, given a hypergraph $H=(X,E)$ and integers $k$ and $\ell$, whether one can select a set $C\subseteq X$ of $k$ vertices of $H$ such that the set $\{e\cap C, e\in E\}$ of distinct hyperedge-intersections with $C$ has size at least $\ell$. The sets $e\cap C$ define equivalence classes over $E$. Partial VC Dimension is a generalization of VC Dimension, which corresponds to the case $\ell=2^k$, and of Distinguishing Transversal, which corresponds to the case $\ell=|E|$ (the latter is also known as Test Cover in the dual hypergraph). We also introduce the associated fixed-cardinality maximization problem Max Partial VC Dimension that aims at maximizing the number of equivalence classes induced by a solution set of $k$ vertices. We study the algorithmic complexity of Partial VC Dimension and Max Partial VC Dimension both on general hypergraphs and on more restricted instances, in particular, neighborhood hypergraphs of graphs.