Shattering, Graph Orientations, and Connectivity

TL;DR

This paper establishes a novel connection between VC-theory and graph theory, deriving new inequalities and providing insights into shattering-extremal systems through graph orientations and connectivity concepts.

Contribution

It introduces a new link between VC-theory and graph theory, generalizes the Sauer-Shelah Lemma, and offers new proofs and examples related to shattering-extremal systems.

Findings

Derived new inequalities for graph orientations.

Provided new proofs for existing graph theory results.

Constructed examples of shattering-extremal systems from network properties.

Abstract

We present a connection between two seemingly disparate fields: VC-theory and graph theory. This connection yields natural correspondences between fundamental concepts in VC-theory, such as shattering and VC-dimension, and well-studied concepts of graph theory related to connectivity, combinatorial optimization, forbidden subgraphs, and others. In one direction, we use this connection to derive results in graph theory. Our main tool is a generalization of the Sauer-Shelah Lemma. Using this tool we obtain a series of inequalities and equalities related to properties of orientations of a graph. Some of these results appear to be new, for others we give new and simple proofs. In the other direction, we present new illustrative examples of shattering-extremal systems - a class of set-systems in VC-theory whose understanding is considered by some authors to be incomplete. These examples are derived from properties of orientations related to distances and flows in networks.