Contradiction Graphs Determine VC Dimension

2026-05-19

AI Summary

This work investigates how the finiteness of the Vapnik–Chervonenkis (VC) dimension of a binary concept class can be determined through graph-theoretic properties. To this end, it introduces the contradiction graph \( G_m(\mathcal{H}) \), whose vertices correspond to length-\( m \) labelings realizable by the hypothesis class \( \mathcal{H} \), with edges connecting labelings that disagree on at least one common input point. The central contribution is the first proof that the structural properties of a single contradiction graph \( G_m(\mathcal{H}) \) are sufficient to decide whether \( \mathrm{VCdim}(\mathcal{H}) \geq m \), thereby fully characterizing the threshold behavior of the VC dimension. This approach establishes an exact correspondence between sequences of contradiction graphs and the VC dimension, effectively resolving a key open question posed by Alon et al. regarding the decidability of VC dimension finiteness.

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).

Problem

• contradiction graphs
• VC dimension
• binary concept classes
• finite vs infinite VC dimension

Innovation

• contradiction graphs
• VC dimension
• binary concept classes
• graph characterization
• combinatorial learning theory