Complexity of computing Vapnik-Chervonenkis dimension

  • Ayumi Shinohara
Selected Papers Approximate Learning
Part of the Lecture Notes in Computer Science book series (LNCS, volume 744)


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(log2J) variables, and (2) for every constant C, the satisfiability problem in conjunctive normal form with m clauses and Clog2m 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 nO(log n time computable sets.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Ayumi Shinohara
    • 1
  1. 1.Research Institute of Fundamental Information ScienceKyushu University 33FukuokaJapan

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