Contradiction Graphs Determine VC Dimension

TL;DR

This paper introduces contradiction graphs for binary concept classes, showing that a single graph can determine the VC dimension, thus providing a new combinatorial tool for understanding model complexity.

Contribution

It proves that the contradiction graph $G_m(H)$ uniquely determines whether the VC dimension of a class is at least m, answering an open question about finite versus infinite VC dimension detection.

Findings

The contradiction graph $G_m(H)$ encodes VC dimension information.

A sequence of contradiction graphs fully determines the VC dimension.

The method distinguishes finite from infinite VC dimension.

Abstract

We study the contradiction graphs associated with binary concept classes. For a class $H \subseteq \{0,1\}^X$, the order-$m$ contradiction graph $G_m(H)$ has as vertices the $H$-realizable labeled sequences of length $m$, with two vertices adjacent when the two sequences assign opposite labels to some common domain point. Our main result is that the single graph $G_m(H)$ determines the threshold predicate $\mathrm{VCdim}(H)\ge m$. Consequently, the full sequence $(G_m(H))_{m \ge 1}$ determines the exact VC dimension and, in particular, detects finite versus infinite VC dimension, answering a question posed by Alon et al. (2024).