Advertisement

On the hardness of approximating the minimum consistent OBDD problem

  • Kouichi Hirata
  • Shinichi Shimozono
  • Ayumi Shinohara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1097)

Abstract

Ordered binary decision diagrams (OBDDs, for short) represent Boolean functions as directed acyclic graphs. The minimum consistent OBDD problem is, given an incomplete truth table of a function, to find the smallest OBDD that is consistent with the truth table with respect to a fixed order of variables. We show that this problem is NP-hard, and prove that there is a constant ε > 0 such that no polynomial time algorithm can approximate the minimum consistent OBDD within the ratio nε unless P=NP, where n is the number of variables. This result suggests that OBDDs are unlikely to be polynomial time learnable in PAC-learning model.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Angluin, D.: On the complexity of minimum inference of regular sets, Information and Control 39, 337–350, 1978.CrossRefGoogle Scholar
  2. 2.
    Angluin, D.: Learning regular sets from queries and counterexamples, Information and Computation 75, 87–106, 1987.CrossRefGoogle Scholar
  3. 3.
    Angluin, D.: Queries and concept learning, Machine Learning 2, 319–342, 1988.Google Scholar
  4. 4.
    Angluin, D.: Negative results for equivalence queries, Machine Learning 5, 121–150, 1990.Google Scholar
  5. 5.
    Blum, A.: New approximation algorithms for graph coloring, Journal of the Association for Computing Machinery 41, 470–516, 1994.Google Scholar
  6. 6.
    Board, R. and Pitt, L.: On the necessity of Occam algorithms, Theoretical Computer Science 100, 157–184, 1992.Google Scholar
  7. 7.
    Bolling, B. and Wegener, I.: Improving the variable ordering of OBDDs is NP-complete, Technical Report, Universität Dortmund, 1994.Google Scholar
  8. 8.
    Bryant, R. E.: Symbolic Boolean manipulation with ordered binary-decision diagrams, ACM Computing Surveys 24, 293–318, 1992.CrossRefGoogle Scholar
  9. 9.
    Ergün, F., Kumar, S. R. and Rubinfeld, R.: On learning bounded-width branching programs, Proc. 8th International Workshop on Computational Learning Theory, 361–368, 1995.Google Scholar
  10. 10.
    Garey, M. and Johnson, D. S.: Computers and intractability: A guide to the theory of NP-completeness, W. H. Freeman and Company, 1978.Google Scholar
  11. 11.
    Gavaldà, R. and Guijarro, D.: Learning ordered binary decision diagrams, Proc. 6th International Workshop on Algorithmic Learning Theory, 228–238, LNAI 997, 1995.Google Scholar
  12. 12.
    Gold, E. M.: Complexity of automaton identification from given data, Information and Control 37, 302–320, 1978.CrossRefGoogle Scholar
  13. 13.
    Hancock, T., Jiang, T., Li, M. and Tromp, J.: Lower bounds on learning decision lists and trees, draft, 1996.Google Scholar
  14. 14.
    Li, M. and Vazirani, U.: On the learnability of finite automata, Proc. 1988 Workshop on Computational Learning Theory, 359–370, 1988.Google Scholar
  15. 15.
    Lund, C. and Yannakakis, M.: On the hardness of approximating minimization problems, Proc. 25th Annual ACM Symposium on Theory of Computing, 286–293, 1993.Google Scholar
  16. 16.
    Pitt, L. and Valiant, L. G.: Computational limitation on learning from examples, Journal of the Association for Computing Machinery 35, 965–984, 1988.Google Scholar
  17. 17.
    Pitt, L. and Warmuth, M. K.: The minimum consistent DFA problem cannot be approximated within any polynomial, Journal of the Association for Computing Machinery 40, 95–142, 1993.Google Scholar
  18. 18.
    Sauerhoff, M. and Wegener, I.: On the complexity of minimizing the OBDD size for incompletely specified functions, Technical Report 560, Univ. Dortmund, 1994.Google Scholar
  19. 19.
    Sieling, D. and Wegener, I.: Reduction of OBDDs in linear time, Information Processing Letter 48, 139–144, 1993.MathSciNetGoogle Scholar
  20. 20.
    Takenaga, Y. and Yajima, S.: NP-completeness of minimum binary decision diagram identification, Technical Report of IEICE, COMP92-99, 57–62, 1993.Google Scholar
  21. 21.
    Valiant, L. G.: A theory of the learnable, Communications of the ACM 27, 1134–1142, 1984.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Kouichi Hirata
    • 1
  • Shinichi Shimozono
    • 2
  • Ayumi Shinohara
    • 3
  1. 1.Department of Artificial IntelligenceKyushu Institute of TechnologyIizukaJapan
  2. 2.Department of Control Engineering and ScienceKyushu Institute of TechnologyIizukaJapan
  3. 3.Research Institute of Fundamental Information ScienceKyushu UniversityFukuokaJapan

Personalised recommendations