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arXiv · 1609.05110

Parameterized and Approximation Complexity of Partial VC Dimension

Cristina Bazgan·Florent Foucaud·Florian Sikora
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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.

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Cristina Bazgan, Florent Foucaud, Florian Sikora. 2016-09-16. Parameterized and Approximation Complexity of Partial VC Dimension. https://doi.org/10.1016/j.tcs.2018.09.013

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