On the Complexity of Approximating the VC Dimension
نویسندگان
چکیده
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 • Σp3-hard to approximate to within a factor 2− for any > 0, • approximable in AM to within a factor 2, and • AM-hard to approximate to within a factor N for some constant > 0. To obtain the Σp3-hardness result we solve a randomness extraction problem using list-decodable binary codes; for the positive result we utilize the Sauer-Shelah(-Perles) Lemma. The exact value of in the AM-hardness result depends on the degree achievable by explicit disperser constructions.
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