# Complexity of computing Vapnik-Chervonenkis dimension

## Abstract

The Vapnik-Chervonenkis (VC) dimension is known to be the crucial measure of the polynomial-sample learnability in the PAC-learning model. This paper investigates the complexity of computing VC-dimension of a concept class over a finite learning domain. We consider a decision problem called the discrete VC-dimension problem which is, for a given matrix representing a concept class F and an integer *K*, to determine whether the VC-dimension of F is greater than *K* or not. We prove that (1) the discrete VC-dimension problem is polynomial-time reducible to the satisfiability problem of length *J* with *O*(log^{2}*J*) variables, and (2) for every constant *C*, the satisfiability problem in conjunctive normal form with *m* clauses and *C*log^{2}*m* variables is polynomial-time reducible to the discrete VC-dimension problem. These results can be interpreted, in some sense, that the problem is “complete” for the class of *n*^{O(log n} time computable sets.

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