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On the complexity of approximating the VC dimension

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2 Author(s)
Mossel, E. ; Microsoft Corp., Redmond, WA, USA ; Umans, C.

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 any ε>0; approximable in Aℳ to within a factor 2; and Aℳ-hard to approximate to within a factor Nε for some constant ε>0. To obtain the Σ39-hardness results we solve a randomness extraction problem using list-decodable binary codes; for the positive results we utilize the Sauer-Shelah(-Perles) Lemma. The exact value of ε in the Aℳ-hardness result depends on the degree achievable by explicit disperser constructions

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Computational Complexity, 16th Annual IEEE Conference on, 2001.

Date of Conference: