Advertisement

On the Isomorphism Problem for Decision Trees and Decision Lists

  • Vikraman Arvind
  • Johannes Köbler
  • Sebastian Kuhnert
  • Gaurav Rattan
  • Yadu Vasudev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8070)

Abstract

We study the complexity of isomorphism testing for Boolean functions that are represented by decision trees or decision lists. Our results include a \(2^{\sqrt{s}(\lg s)^{O(1)}}\) time algorithm for isomorphism testing of decision trees of size s. Additionally, we show:
  • Isomorphism testing of rank-1 decision trees is complete for logspace.

  • For r ≥ 2, isomorphism testing for rank-r decision trees is polynomial-time equivalent to Graph Isomorphism. As a consequence we obtain a \({2^{\sqrt{s}(\lg s)^{O(1)}}}\) time algorithm for isomorphism testing of decision trees of size s.

  • The isomorphism problem for decision lists admits a Schaefer-type dichotomy: depending on the class of base functions, the isomorphism problem is either in polynomial time, or equivalent to Graph Isomorphism, or coNP-hard.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ADKK12]
    Arvind, V., Das, B., Köbler, J., Kuhnert, S.: The isomorphism problem for k-trees is complete for logspace. Inf. Comput. 217, 1–11 (2012)zbMATHCrossRefGoogle Scholar
  2. [AT96]
    Agrawal, M., Thierauf, T.: The Boolean Isomorphism Problem. In: FOCS, pp. 422–430 (1996)Google Scholar
  3. [BC08]
    Babai, L., Codenotti, P.: Isomorhism of hypergraphs of low rank in moderately exponential time. In: FOCS, pp. 667–676 (2008)Google Scholar
  4. [BHRV04]
    Böhler, E., Hemaspaandra, E., Reith, S., Vollmer, H.: The complexity of Boolean constraint isomorphism. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 164–175. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. [BL83]
    Babai, L., Luks, E.M.: Canonical labeling of graphs. In: STOC, pp. 171–183 (1983)Google Scholar
  6. [EH89]
    Ehrenfeucht, A., Haussler, D.: Learning decision trees from random examples. Inf. Comput. 82(3), 231–246 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Luk99]
    Luks, E.M.: Hypergraph isomorphism and structural equivalence of Boolean functions. In: STOC, pp. 652–658 (1999)Google Scholar
  8. [Riv87]
    Rivest, R.L.: Learning decision lists. Machine Learning 2(3), 229–246 (1987)Google Scholar
  9. [RZ00]
    Ramnath, S., Zhao, P.: On the isomorphism of expressions. Inf. Process. Lett. 74(3-4), 97–102 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [Sch78]
    Schaefer, T.J.: The complexity of satisfiability problems. In: STOC, pp. 216–226 (1978)Google Scholar
  11. [Thi00]
    Thierauf, T.: The Computational Complexity of Equivalence and Isomorphism Problems. LNCS, vol. 1852. Springer (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Vikraman Arvind
    • 1
  • Johannes Köbler
    • 2
  • Sebastian Kuhnert
    • 2
  • Gaurav Rattan
    • 1
  • Yadu Vasudev
    • 1
  1. 1.The Institute of Mathematical SciencesChennaiIndia
  2. 2.Institut für InformatikHumboldt-Universität zu BerlinGermany

Personalised recommendations