skip to main content
10.1145/2463664.2463673acmconferencesArticle/Chapter ViewAbstractPublication PagesmodConference Proceedingsconference-collections
research-article

Deciding monotone duality and identifying frequent itemsets in quadratic logspace

Published:22 June 2013Publication History

ABSTRACT

The monotone duality problem is defined as follows: Given two monotone formulas f and g in irredundant DNF, decide whether f and g are dual. This problem is the same as duality testing for hypergraphs, that is, checking whether a hypergraph H consists of precisely all minimal transversals of a hypergraph G. By exploiting a recent problem-decomposition method by Boros and Makino (ICALP 2009), we show that duality testing for hypergraphs, and thus for monotone DNFs, is feasible in DSPACE(log2 n), i.e., in quadratic logspace. As the monotone duality problem is equivalent to a number of problems in the areas of databases, data mining, and knowledge discovery, the results presented here yield new complexity results for those problems, too. For example, it follows from our results that whenever, for a Boolean-valued relation (whose attributes represent items), a number of maximal frequent itemsets and a number of minimal infrequent itemsets are known, then it can be decided in quadratic logspace whether there exist additional frequent or infrequent itemsets.

References

  1. I. Adler, G. Gottlob, and M. Grohe. Hypertree width and related hypergraph invariants. Eur. J. Comb., 28(8):2167--2181, 2007. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. E. Boros, V. Gurvich, L. Khachiyan, and K. Makino. On the complexity of generating maximal frequent and minimal infrequent sets. In Proceedings 19th Symposium on Theoretical Aspects of Computing (STACS'02), number 2285 in LNCS, pages 133--141, 2002. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. E. Boros, V. Gurvich, L. Khachiyan, and K. Makino. On maximal frequent and minimal infrequent sets in binary matrices. Ann. Math. Artif. Intell., 39(3):211--221, 2003. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. E. Boros and K. Makino. A fast and simple parallel algorithm for the monotone duality problem. In S. Albers, A. Marchetti-Spaccamela, Y. Matias, S. E. Nikoletseas, and W. Thomas, editors, ICALP (Part I), volume 5555 of Lecture Notes in Computer Science, pages 183--194. Springer, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. L. Cai and J. Chen. On the amount of nondeterminism and the power of verifying (extended abstract). In A. M. Borzyszkowski and S. Sokolowski, editors, MFCS, volume 711 of Lecture Notes in Computer Science, pages 311--320. Springer, 1993. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. J. Demetrovics and V. D. Thi. Armstrong relations, functional dependencies and strong dependencies. Computers and Artificial Intelligence, 14(3), 1995.Google ScholarGoogle Scholar
  7. T. Eiter and G. Gottlob. Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput., 24(6):1278--1304, 1995. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. T. Eiter and G. Gottlob. Hypergraph transversal computation and related problems in Logic and AI. In S. Flesca, S. Greco, N. Leone, and G. Ianni, editors, JELIA, volume 2424 of Lecture Notes in Computer Science, pages 549--564. Springer, 2002. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. T. Eiter, G. Gottlob, and K. Makino. New results on monotone dualization and generating hypergraph transversals. SIAM J. Comput., 32(2):514--537, 2003. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. T. Eiter and K. Makino. Generating all abductive explanations for queries on propositional horn theories. In M. Baaz and J. A. Makowsky, editors, CSL, volume 2803 of Lecture Notes in Computer Science, pages 197--211. Springer, 2003.Google ScholarGoogle Scholar
  11. T. Eiter, K. Makino, and G. Gottlob. Computational aspects of monotone dualization: A brief survey. Discrete Applied Mathematics, 156(11):2035--2049, 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. K. M. Elbassioni. On the complexity of monotone dualization and generating minimal hypergraph transversals. Discrete Applied Mathematics, 156(11):2109--2123, 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. E. A. Emerson and C. S. Jutla. Tree automata, mu-calculus and determinacy (extended abstract). In FOCS, pages 368--377. IEEE Computer Society, 1991. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. E. A. Emerson, C. S. Jutla, and A. P. Sistla. On model-checking for fragments of μ-calculus. In C. Courcoubetis, editor, CAV, volume 697 of Lecture Notes in Computer Science, pages 385--396. Springer, 1993. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. M. Fredman and L. Khachiyan. On the Complexity of Dualization of Monotone Disjunctive Normal Forms. Journal of Algorithms, 21:618--628, 1996. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. H. Garcia-Molina and D. Barbará. How to assign votes in a distributed system. J. ACM, 32(4):841--860, 1985. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. D. Gaur. Satisfiability and self-duality of monotone Boolean functions. Dissertation, School of Computing Science, Simon Fraser University, January 1999. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. D. Gaur and R. Krishnamurti. Self-duality of bounded monotone boolean functions and related problems. In Proceedings 11th International Conference on Algorithmic Learning Theory (ALT), number 1968 in LNCS, pages 209--223, 2000. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. G. Gogic, C. Papadimitriou, and M. Sideri. Incremental Recompilation of Knowledge. Journal of Artificial Intelligence Research, 8:23--37, 1998. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. J. Goldsmith, M. A. Levy, and M. Mundhenk. Limited nondeterminism. SIGACT News, 27(2):20--29, 1996. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. G. Gottlob, N. Leone, and F. Scarcello. Hypertree decompositions: A survey. In J. Sgall, A. Pultr, and P. Kolman, editors, MFCS, volume 2136 of Lecture Notes in Computer Science, pages 37--57. Springer, 2001. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. G. Gottlob, N. Leone, and F. Scarcello. Hypertree decompositions and tractable queries. J. Comput. Syst. Sci., 64(3):579--627, 2002.Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. G. Gottlob and L. Libkin. Investigations on Armstrong Relations, Dependency Inference, and Excluded Functional Dependencies. Acta Cybernetica, 9(4):385--402, 1990.Google ScholarGoogle Scholar
  24. R. Greiner, B. A. Smith, and R. W. Wilkerson. A Correction to the Algorithm in Reiter's Theory of Diagnosis. Artificial Intelligence, 41:79--88, 1990. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. D. Gunopulos, R. Khardon, H. Mannila, S. Saluja, H. Toivonen, and R. S. Sharm. Discovering all most specific sentences. ACM Trans. Database Syst., 28(2):140--174, 2003. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. D. Gunopulos, R. Khardon, H. Mannila, and H. Toivonen. Data mining, hypergraph transversals, and machine learning. In Proceedings of the 16th ACM SIGACT SIGMOD-SIGART Symposium on Principles of Database Systems (PODS-96), pages 209--216, 1993. Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. M. Hagen. Algorithmic and Computational Complexity Issues of MONET. Cuvillier Verlag, 2008.Google ScholarGoogle Scholar
  28. T. Harada and M. Yamashita. Transversal merge operation: a nondominated coterie construction method for distributed mutual exclusion. IEEE Transactions on Parallel and Distributed Systems, 16(2):183--192, Feb. Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. T. A. Henzinger, O. Kupferman, and R. Majumdar. On the universal and existential fragments of the μ-calculus. In H. Garavel and J. Hatcliff, editors, TACAS, volume 2619 of Lecture Notes in Computer Science, pages 49--64. Springer, 2003. Google ScholarGoogle ScholarDigital LibraryDigital Library
  30. T. Ibaraki and T. Kameda. A theory of coteries: Mutual exclusion in distributed systems. IEEE Trans. Parallel Distrib. Syst., 4(7):779--794, 1993. Google ScholarGoogle ScholarDigital LibraryDigital Library
  31. D. S. Johnson. A Catalog of Complexity Classes. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume A, chapter 2. Elsevier Science Publishers B.V. (North-Holland), 1990. Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. M. Jurdzinski. Deciding the winner in parity games is in upnco-up. Inf. Process. Lett., 68(3):119--124, 1998. Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. D. Kavvadias, C. Papadimitriou, and M. Sideri. On Horn Envelopes and Hypergraph Transversals. In W. Ng, editor, Proceedings 4th International Symposium on Algorithms and Computation (ISAAC-93), LNCS 762, pages 399--405, Hong Kong, December 1993. Springer. Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. D. J. Kavvadias and E. C. Stavropoulos. Monotone Boolean Dualization is in co-NP{log$^2$n}. Information Processing Letters, 85:1--6, 2003. Google ScholarGoogle ScholarDigital LibraryDigital Library
  35. L. Lamport. The implementation of reliable distributed multiprocess systems. Comp. Networks, 2:95--114, 1978.Google ScholarGoogle Scholar
  36. K. K. Loo, C. L. Yip, B. Kao, and D. W.-L. Cheung. Exploiting the duality of maximal frequent itemsets and minimal infrequent itemsets for i/o efficient association rule mining. In M. T. Ibrahim, J. Küng, and N. Revell, editors, DEXA, volume 1873 of Lecture Notes in Computer Science, pages 710--719. Springer, 2000. Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. K. Makino and T. Kameda. Efficient generation of all regular non-dominated coteries. In G. Neiger, editor, PODC, pages 279--288. ACM, 2000. Google ScholarGoogle ScholarDigital LibraryDigital Library
  38. K. Makino and T. Kameda. Transformations on regular nondominated coteries and their applications. SIAM J. Discrete Math., 14(3):381--407, 2001. Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. H. Mannila and H. Toivonen. Levelwise search and borders of theories in knowledge discovery. Data mining and knowledge discovery, 1(3):241--258, 1997. Google ScholarGoogle ScholarDigital LibraryDigital Library
  40. C. H. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.Google ScholarGoogle Scholar
  41. R. Reiter. A theory of diagnosis from first principles. Artif. Intell., 32(1):57--95, 1987. Google ScholarGoogle ScholarDigital LibraryDigital Library
  42. D. Saccà and E. Serra. Number of minimal hypergraph transversals and complexity of inverse frequent itemsets mining with infrequency constraints: they are high in theory, but often not so much in practice! Manuscript, currently unpublished, 2013.Google ScholarGoogle Scholar
  43. K. Satoh and T. Uno. Enumerating maximal frequent sets using irredundant dualization. In G. Grieser, Y. Tanaka, and A. Yamamoto, editors, Discovery Science, volume 2843 of Lecture Notes in Computer Science, pages 256--268. Springer, 2003.Google ScholarGoogle Scholar
  44. H. Tamaki. Space-efficient enumeration of minimal transversals of a hypergraph. IPSJ-AL, 75:29--36, 2000.Google ScholarGoogle Scholar
  45. W. Zielonka. Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theor. Comput. Sci., 200(1--2):135--183, 1998. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Deciding monotone duality and identifying frequent itemsets in quadratic logspace

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in
      • Published in

        cover image ACM Conferences
        PODS '13: Proceedings of the 32nd ACM SIGMOD-SIGACT-SIGAI symposium on Principles of database systems
        June 2013
        334 pages
        ISBN:9781450320665
        DOI:10.1145/2463664
        • General Chair:
        • Richard Hull,
        • Program Chair:
        • Wenfei Fan

        Copyright © 2013 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 22 June 2013

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • research-article

        Acceptance Rates

        PODS '13 Paper Acceptance Rate24of97submissions,25%Overall Acceptance Rate476of1,835submissions,26%

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader
      About Cookies On This Site

      We use cookies to ensure that we give you the best experience on our website.

      Learn more

      Got it!