On the complexity of approximating the VC dimension

Authors:
Elchanan Mossel;Christopher Umans
Affiliations:
Microsoft Research, One Microsoft Way, Redmond WA;Microsoft Research, One Microsoft Way, Redmond WA
Venue:
Journal of Computer and System Sciences - Complexity 2001
Year:
2002

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Abstract

We study the complexity of approximating the VC dimension of a collection of sets, when the sets are encoded succinctly by a small circuit We show that this problem is: • Σ3p-hard to approximate to within a factor 2 -ε for all ε A M to within a factor 2, and • A M-hard to approximate to within a factor N1-ε for all ε Σ3p-hardness result we solve a randomness extraction problem using list-decodable binary codes; for the positive result we utilize the Sauer-Shelah(-Perles) Lemma. We prove analogous results for the q-ary VC dimension, where the approximation threshold is q.