Complexity of computing Vapnik-Chervonenkis dimension and some generalized dimensions

Citations of this article
Mendeley users who have this article in their library.


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 ΨG-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 ΨG-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. © 1995.




Shinohara, A. (1995). Complexity of computing Vapnik-Chervonenkis dimension and some generalized dimensions. Theoretical Computer Science, 137(1), 129–144.

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free