On the complexity of constrained VC-classes

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Sauer's lemma is extended to classes HN of binary-valued functions h on [n] = { 1, ..., n } which have a margin less than or equal to N on all x ∈ [n] with h (x) = 1, where the margin μh (x) of h at x ∈ [n] is defined as the largest non-negative integer a such that h is constant on the interval Ia (x) = [x - a, x + a] ⊆ [n]. Estimates are obtained for the cardinality of classes of binary-valued functions with a margin of at least N on a positive sample S ⊆ [n].

Original languageEnglish
Pages (from-to)903-910
Number of pages8
JournalDiscrete Applied Mathematics
Issue number6
StatePublished - 15 Mar 2008
Externally publishedYes


  • Boolean functions
  • Integer partitions
  • Sauer's lemma
  • VC-dimension


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