Complexity of computing Vapnik-Chervonenkis dimension and some generalized dimensions

In the PAC-learning model, the Vapnik-Chervonenkis (VC) dimension plays the key role to estimate the polynomial-sample learnability of a class of {0, 1}-valued functions. For a class of {0, ..., N}-valued functions, the notion has been generalized in various ways. This paper investigates the complexity of computing VC-dimension and generalized dimensions: VC^*-dimension, Ψ_*-dimension, and Ψ_G-dimension. For each dimension, we consider a decision problem that is, for a given matrix representing a class F of functions and an integer K, to determine whether the dimension of F is greater than K or not. We prove that (1) both the VC^*-dimension and Ψ_G-dimension problems are polynomial-time reducible to the satisfiability problem of length J with O(log² J) variables, which include the original VC-dimension problem as a special case, (2) for every constant C, the satisfiability problem in conjunctive normal form with m clauses and C log² m variables is polynomial-time reducible to the VC-dimension problem, while (3) Ψ_*-dimension problem is NP-complete.