SIGACT news complexity theory comun 37
ACM SIGACT News
SIGACT news complexity theory column 38
ACM SIGACT News
Increasing the Output Length of Zero-Error Dispersers
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
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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 1) \Sigma^p_3-hard to approximate to within a factor 2 -\epsilon for any \epsilon 0, 2) approximable in {\cal AM} to within a factor 2 , and 3) {\cal AM}-hard to approximate to within a factor N^\epsilon for some constant \epsilon 0. To obtain the \Sigma^p_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 \epsilon in the {\cal AM}-hardness result depends on the degree achievable by explicit disperser constructions.