Contradiction Graphs Determine VC Dimension
Abstract
We study the contradiction graphs associated with binary concept classes. For a class H ⊂eq \0,1\X, the order-m contradiction graph Gm(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 Gm(H) determines the threshold predicate VCdim(H) m. Consequently, the full sequence (Gm(H))m 1 determines the exact VC dimension and, in particular, detects finite versus infinite VC dimension, answering a question posed by Alon et al. (2024).
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