On the Complexity of Approximating the VC Dimension

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 • Σp 3-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 Σp 3-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.

• Type: Conference
• Year: 2001
• Where: COCO
• Authors: Elchanan Mossel, Christopher Umans
• Keywords: Algorithms, COCO 2001, Factor 2, Randomness Extraction Problem, VC Dimension