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: /spl Sigma//sub 3//sup p/-hard to approximate to within a factor 2-/spl epsiv/ for any /spl epsiv/>0; approximable in A/spl Mscr/ to within a factor 2; and A/spl Mscr/-hard to approximate to within a factor N/sup /spl epsiv// for some constant /spl epsiv/>0. To obtain the /spl Sigma//sub 3//sup 9/-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 /spl epsiv/ in the A/spl Mscr/-hardness result depends on the degree achievable by explicit disperser constructions.