Hypergraph Transversal Computation and Related Problems in Logic and AI

  • Thomas Eiter
  • Georg Gottlob
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2424)


Generating minimal transversals of a hypergraph is an important problem which has many applications in Computer Science. In the present paper, we address this problem and its decisional variant, i.e., the recognition of the transversal hypergraph for another hypergraph. We survey some results on problems which are known to be related to computing the transversal hypergraph, where we focus on problems in propositional Logic and AI. Some of the results have been established already some time ago, and were announced but their derivation was not widely disseminated. We then address recent developments on the computational complexity of computing resp. recognizing the transversal hypergraph. The precise complexity of these problems is not known to date, and is in fact open for more than 20 years now.


Polynomial Time Boolean Function Conjunctive Query Membership Query Negative Clause 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    C. Alchourrón, P. Gärdenfors, and D. Makinson. On the logic of theory change: Partial meet contraction and revision functions. J. Symb. Logic, 50:510–530, 1985.zbMATHCrossRefGoogle Scholar
  2. 2.
    C. Benzaken. Algorithme de dualisation d’une fonction booléenne. Revue Francaise de Traitment de l’Information-Chiffres, 9(2):119–128, 1966.MathSciNetGoogle Scholar
  3. 3.
    C. Bioch and T. Ibaraki. Complexity of identification and dualization of positive Boolean functions. Information and Computation, 123:50–63, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    E. Boros, K. Elbassioni, V. Gurvich, and L. Khachiyan. An efficient incremental algorithm for generating all maximal independent sets in hypergraphs of bounded dimension. Parallel Processing Letters, 10(4):253–266, 2000.CrossRefMathSciNetGoogle Scholar
  5. 5.
    E. Boros, V. Gurvich, and P. L. Hammer. Dual subimplicants of positive Boolean functions. Optimization Methods and Software, 10:147–156, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    E. Boros, V. Gurvich, L. Khachiyan, and K. Makino. On the complexity of generating maximal frequent and minimal infrequent sets. In Proc. STACS-02, LNCS 2285, pp. 133–141, 2002.Google Scholar
  7. 7.
    E. Boros, P. Hammer, T. Ibaraki, and K. Kawakami. Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle. SIAM J. Comput., 26(1):93–109, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. de Kleer and B. C. Williams. Diagnosing multiple faults. Artificial Intelligence, 32:97–130, 1987.zbMATHCrossRefGoogle Scholar
  9. 9.
    C. Domingo, N. Mishra, and L. Pitt. Eficient read-restricted monotone CNF/DNF dualization by learning with membership queries. Machine Learning, 37:89–110, 1999.zbMATHCrossRefGoogle Scholar
  10. 10.
    T. Eiter. On Transversal Hypergraph Computation and Deciding Hypergraph Saturation. PhD thesis, Institut für Informationssysteme, TU Wien, Austria, 1991.Google Scholar
  11. 11.
    T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing, 24(6):1278–1304, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    T. Eiter, G. Gottlob, and K. Makino. New results on monotone dualization and generating hypergraph transversals. In Proc. ACM STOC-2002, pp. 14–22, 2002. Full paper Tech. Rep. INFSYS RR-1843-02-05, TU Wien. Available as Computer Science Repository Report (CoRR) nr. cs.DS/0204009 via URL:
  13. 13.
    T. Eiter and K. Makino. On computing all abductive explanations. In Proc. 18th National Conference on Artificial Intelligence (AAAI’ 02). AAAI Press, 2002.Google Scholar
  14. 14.
    T. Eiter, K. Makino, and T. Ibaraki. Decision lists and related Boolean functions. Theoretical Computer Science, 270(1–2):493–524, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    R. Fagin. Degrees of acyclicity for hypergraphs and relational database schemes. Journal of the ACM, 30:514–550, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    R. Fagin, J. D. Ullman, and M. Y. Vardi. On the semantics of updates in databases. In Proc. PODS-83, pp. 352–365, 1983.Google Scholar
  17. 17.
    M. Fredman and L. Khachiyan. On the complexity of dualization of monotone disjunctive normal forms. Journal of Algorithms, 21:618–628, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    G. Friedrich, G. Gottlob, and W. Nejdl. Physical negation instead of fault models. In Proc. AAAI-91, July 1990.Google Scholar
  19. 19.
    P. Gärdenfors. Knowledge in Flux. Bradford Books, MIT Press, 1988.Google Scholar
  20. 20.
    M. Garey and D. S. Johnson. Computers and Intractability-A Guide to the Theory of NP-Completeness. W. H. Freeman, New York, 1979.zbMATHGoogle Scholar
  21. 21.
    D. Gaur and R. Krishnamurti. Self-duality of bounded monotone Boolean functions and related problems. In Proc. 11th International Conference on Algorithmic Learning Theory (ALT), LNCS 1968, pp. 209–223. Springer, 2000.Google Scholar
  22. 22.
    M. L. Ginsberg. Counterfactuals. Artificial Intelligence, 30:35–79, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    M. L. Ginsberg and D. E. Smith. Reasoning about action I: A possible worlds approach. Artificial Intelligence, 35:165–195, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    G. Gogic, C. Papadimitriou, and M. Sideri. Incremental recompilation of knowledge. J. Artificial Intelligence Research, 8:23–37, 1998.zbMATHMathSciNetGoogle Scholar
  25. 25.
    J. Goldsmith, M. Levy, and M. Mundhenk. Limited nondeterminism. SIGACT News, 27(2):20–29, 1978.CrossRefGoogle Scholar
  26. 26.
    G. Gottlob, N. Leone, and F. Scarcello. Hypertree decompositions and tractable queries. In Proc. 18th ACM Symp. on Principles of Database Systems (PODS-99), pp. 21–32, 1999. Full paper to appear in Journal of Computer and System Sciences.Google Scholar
  27. 27.
    G. Gottlob and L. Libkin. Investigations on Armstrong relations, dependency inference, and excluded functional dependencies. Acta Cybernetica, 9(4):385–402, 1990.zbMATHMathSciNetGoogle Scholar
  28. 28.
    D. Gunopulos, R. Khardon, H. Mannila, and H. Toivonen. Data mining, hypergraph transversals, and machine learning. In Proc. 16th ACM Symp. on Principles of Database Systems (PODS-97), pp. 209–216, 1997.Google Scholar
  29. 29.
    D. S. Johnson. A Catalog of Complexity Classes. In J. van Leeuwen, ed., Handbook of Theoretical Computer Science, A, chapter 2. Elsevier, 1990.Google Scholar
  30. 30.
    D. S. Johnson, M. Yannakakis, and C. H. Papadimitriou. On generating all maximal independent sets. Information Processing Letters, 27:119–123, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    H. Kautz, M. Kearns, and B. Selman. Reasoning with characteristic models. In Proc. AAAI-93, pp. 34–39, 1993.Google Scholar
  32. 32.
    D. Kavvadias, C. Papadimitriou, and M. Sideri. On Horn envelopes and hypergraph transversals. In W. Ng, editor, Proc. 4th International Symposium on Algorithms and Computation (ISAAC-93), LNCS 762, pp. 399–405, 1993.Google Scholar
  33. 33.
    R. Khardon. Translating between Horn representations and their characteristic models. J. Artificial Intelligence Research, 3:349–372, 1995.zbMATHGoogle Scholar
  34. 34.
    N. Linial and M. Tarsi. Deciding hypergraph 2-colorability by H-resolution. Theoretical Computer Science, 38:343–347, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    K. Makino and T. Ibaraki. A fast and simple algorithm for identifying 2-monotonic positive Boolean functions. Journal of Algorithms, 26:291–302, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    H. Mannila and K.-J. Räihä. Design by Example: An application of Armstrong relations. Journal of Computer and System Sciences, 22(2):126–141, 1986.CrossRefGoogle Scholar
  37. 37.
    N. Mishra and L. Pitt. Generating all maximal independent sets of boundeddegree hypergraphs. In Proc. Tenth Annual Conference on Computational Learning Theory (COLT-97), pp. 211–217, 1997.Google Scholar
  38. 38.
    B. Nebel. A knowledge level analysis of belief revision. In Proc. 1st Intl. Conf. on Principles of Knowledge Representation and Reasoning (KR-89), pp. 301–311, 1989.Google Scholar
  39. 39.
    B. Nebel. How Hard is it to Revise a Belief Base? In D. Gabbay and Ph. Smets, eds, Handbook on Defeasible Reasoning and Uncertainty Management Systems, volume III: Belief Change, pp. 77–145. Kluwer Academic, 1998.Google Scholar
  40. 40.
    C. H. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.Google Scholar
  41. 41.
    R. Reiter. A theory of diagnosis from first principles. Artificial Intelligence, 32:57–95, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    B. Selman and H. J. Levesque. Abductive and default reasoning: A computational core. In Proc. AAAI-90, pp. 343–348, 1990.Google Scholar
  43. 43.
    I. Shmulevich, A. Korshunov, and J. Astola. Almost all monotone boolean functions are polynomially learnable using membership queries. Information Processing Letters, 79:211–213, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    K. Takata. On the sequential method for listing minimal hitting sets. In Proc. Workshop on Discrete Mathematics and Data Mining, 2nd SIAM International Conference on Data Mining, April 11–13, Arlington, Virginia, USA, 2002.Google Scholar
  45. 45.
    M. Winslett. Updating Logical Databases. Cambridge University Press, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Thomas Eiter
    • 1
  • Georg Gottlob
    • 1
  1. 1.Institut für InformationssystemeTechnische Universität WienWienAustria

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