Date of this Version
Journal of Computer and System Sciences
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>0,
- •approximable in AM to within a factor 2, and
- •AM-hard to approximate to within a factor N1−ε for all ε>0.
To obtain the 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.
© 2002. This manuscript version is made available under the CC-BY-NC-ND 4.0 license.
Mossel, E., & Umans, C. (2002). On the Complexity of Approximating the VC Dimension. Journal of Computer and System Sciences, 65 (4), 660-671. http://dx.doi.org/10.1016/S0022-0000(02)00022-3
Date Posted: 27 November 2017
This document has been peer reviewed.