Complexity of Computing Various Generalized VC-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 binary functions. For a class of multi-valued functions, the notion has been generalized in various ways. This paper investigates the complexity of computing some of generalized VC-dimensions: VC*-dimension, *,-dimension, and SG-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 the VC*-dimension problem is polynomial-time reducible to the satisfiability problem of length J with 0 (log2 J) variables, which includes the original VC-dimension problem as a special case. We also show that the qG-dimension problem is still reducible to the satisfiability problem of length J with 0(log2 J) , while the *,-dimension problem becomes NP-complete.