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Generating Partial and Multiple Transversals of a Hypergraph

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Automata, Languages and Programming (ICALP 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1853))

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Abstract

We consider two natural generalizations of the notion of transversal to a finite hypergraph, arising in data-mining and machine learning, the so called multiple and partial transversals. We show that the hypergraphs of all multiple and all partial transversals are dual-bounded in the sense that in both cases, the size of the dual hypergraph is bounded by a polynomial in the cardinality and the length of description of the input hypergraph. Our bounds are based on new inequalities of extremal set theory and threshold logic, which may be of independent interest. We also show that the problems of generating all multiple and all partial transversals of an arbitrary hypergraph are polynomial-time reducible to the well-known dualization problem of hypergraphs. As a corollary, we obtain incremental quasi-polynomial-time algorithms for both of the above problems, as well as for the generation of all the minimal Boolean solutions for an arbitrary monotone system of linear inequalities. Thus, it is unlikely that these problems are NP-hard.

The research of the first two authors was supported in part by the Office of Naval Research (Grant N00014-92-J-1375), the National Science Foundation (Grant DMS 98-06389), and DIMACS. The research of the third author was supported in part by the National Science Foundation (Grant CCR-9618796). The authors are also thankful to József Beck for helpful discussions.

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References

  1. R. Agrawal, H. Mannila, R. Srikant, H. Toivonen and A. I. Verkamo, Fast discovery of association rules, In U. M. Fayyad, G. Piatetsky-Shapiro, P. Smyth and R. Uthurusamy eds., Advances in Knowledge Discovery and Data Mining, 307–328, AAAI Press, Menlo Park, California, 1996.

    Google Scholar 

  2. M. Anthony and N. Biggs, Computational Learning Theory, Cambridge University Press, 1992.

    Google Scholar 

  3. P. Bertolazzi and A. Sassano, An O(mn) time algorithm for regular set-covering problems, Theoretical Computer Science 54 (1987) 237–247.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. C. Bioch and T. Ibaraki, Complexity of identification and dualization of positive Boolean functions, Information and Computation 123 (1995) 50–63.

    Article  MATH  MathSciNet  Google Scholar 

  5. E. Boros, V. Gurvich, and P.L. Hammer, Dual subimplicants of positive Boolean functions, Optimization Methods and Software, 10 (1998) 147–156.

    Article  MATH  MathSciNet  Google Scholar 

  6. E. Boros, V. Gurvich, L. Khachiyan and K. Makino, Dual-Bounded Hypergraphs: Generating Partial and Multiple Transversals, DIMACS Technical Report 99-62, Rutgers University, 1999. (http://dimacs.rutgers.edu/TechnicalReports/1999.html)

  7. E. Boros, P. L. Hammer, T. Ibaraki and K. Kawakami, Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle, SIAM Journal on Computing 26 (1997) 93–109.

    Article  MATH  MathSciNet  Google Scholar 

  8. C. J. Colbourn, The combinatorics of network reliability, Oxford University Press, 1987.

    Google Scholar 

  9. Y. Crama, Dualization of regular Boolean functions, Discrete Applied Mathematics, 16 (1987) 79–85.

    Article  MATH  MathSciNet  Google Scholar 

  10. T. Eiter and G. Gottlob, Identifying the minimal transversals of a hypergraph and related problems, SIAM Journal on Computing, 24 (1995) 1278–1304.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. L. Fredman and L. Khachiyan, On the complexity of dualization of monotone disjunctive normal forms. Journal of Algorithms, 21 (1996) 618–628.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. R. Garey and D. S. Johnson, Computers and Intractability, Freeman, New York, 1979.

    MATH  Google Scholar 

  13. V. Gurvich, To theory of multistep games, USSR Comput. Math. and Math Phys. 13–6 (1973) 1485–1500.

    Google Scholar 

  14. V. Gurvich, Nash-solvability of games in pure strategies, USSR Comput. Math and Math. Phys. 15–2 (1975) 357–371.

    Google Scholar 

  15. V. Gurvich and L. Khachiyan, On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions, Discrete Applied Mathematics, 97 (1999) 363–373.

    Article  MathSciNet  Google Scholar 

  16. V. Gurvich and L. Khachiyan, On the frequency of the most frequently occurring variable in dual DNFs, Discrete Mathematics 169 (1997) 245–248.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. R. Jerrum, L. G. Valiant and V. V. Vazirani, Random generation of combinatorial structures from a uniform distribution Theoretical Computer Science 43 (1986) 169–188.

    Article  MATH  MathSciNet  Google Scholar 

  18. D. S. Johnson, M. Yannakakis and C. H. Papadimitriou, On generating all maximal independent sets, Information Processing Letters, 27 (1988) 119–123.

    Article  MATH  MathSciNet  Google Scholar 

  19. R. Karp and M, Luby, Monte-Carlo algorithms for enumeration and reliability problems, in Proc. 24th IEEE Symp. on Foundations of Computer Science (1983) 56–64.

    Google Scholar 

  20. E. Lawler, J. K. Lenstra and A. H. G. Rinnooy Kan, Generating all maximal independent sets: NP-hardness and polynomial-time algorithms, SIAM Journal on Computing, 9 (1980) 558–565.

    Article  MATH  MathSciNet  Google Scholar 

  21. K. Makino and T. Ibaraki, Interor and exterior functions of Boolean functions, Discrete Applied Mathematics, 69 (1996) 209–231.

    Article  MATH  MathSciNet  Google Scholar 

  22. K. Makino and T. Ibaraki, Inner-core and outer-core functions of partially defined Boolean functions, Discrete Applied Mathematics, 96–97 (1999), 307–326.

    MathSciNet  Google Scholar 

  23. K. Makino and T. Ibaraki, A fast and simple algorithm for identifying 2-monotonic positive Boolean functions, Journal of Algorithms, 26 (1998) 291–305.

    Article  MATH  MathSciNet  Google Scholar 

  24. H. Mannila and K. J. Räihä, Design by example: An application of Armstrong relations, Journal of Computer and System Science 22 (1986) 126–141.

    Article  Google Scholar 

  25. B. Morris and A. Sinclair, Random walks on truncated cubes and sampling 0-1 knapsack problem, in Proc. 40th IEEE Symp. on Foundations of Computer Science (1999) 230–240.

    Google Scholar 

  26. U. N. Peled and B. Simeone, Polynomial-time algorithm for regular set-covering and threshold synthesis, Discrete Applied Mathematics 12 (1985) 57–69.

    Article  MATH  MathSciNet  Google Scholar 

  27. U. N. Peled and B. Simeone, An O(nm)-time algorithm for computing the dual of a regular Boolean function, Discrete Applied Mathematics 49 (1994) 309–323.

    Article  MATH  MathSciNet  Google Scholar 

  28. K. G. Ramamurthy, Coherent Structures and Simple Games, Kluwer Academic Publishers, 1990.

    Google Scholar 

  29. R. C. Read, Every one a winner, or how to avoid isomorphism when cataloging combinatorial configurations, Annals of Discrete Mathematics 2 (1978) 107–120.

    Article  MATH  MathSciNet  Google Scholar 

  30. R. C. Read and R. E. Tarjan, Bounds on backtrack algorithms for listing cycles, paths, and spanning trees, Networks 5 (1975) 237–252.

    MATH  MathSciNet  Google Scholar 

  31. R. H. Sloan, K. Takata, G. Turan, On frequent sets of Boolean matrices, Annals of Mathematics and Artificial Intelligence 24 (1998) 1–4.

    Article  MathSciNet  Google Scholar 

  32. S. Tsukiyama, M. Ide, H. Ariyoshi and I. Shirakawa, A new algorithm for generating all maximal independent sets, SIAM Journal on Computing, 6 (1977) 505–517.

    Article  MATH  MathSciNet  Google Scholar 

  33. J. D. Ullman, Principles of Database and Knowledge Base Systems, Vols. 1 and 2, Computer Science Press, 1988.

    Google Scholar 

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Author information

Authors and Affiliations

  1. RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, NJ, 08854

    Endre Boros

  2. Russian Academy of Sciences, Moscow, Russia

    Vladimir Gurvich

  3. Department of Computer Science, Rutgers University, New Brunswick, New Jersey, 08903

    Leonid Khachiyan

  4. Department of Systems and Human Science, Graduate School of EngineeringScience, Osaka University, Toyonaka, Osaka, 560-8531, Japan

    Kazuhisa Makino

Authors
  1. Endre Boros
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  2. Vladimir Gurvich
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  3. Leonid Khachiyan
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  4. Kazuhisa Makino
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Editor information

Editors and Affiliations

  1. Department of Computer Sciences, University of Pisa, Corso Italia, 40, 56125, Pisa, Italy

    Ugo Montanari

  2. Center for Computer Sciences, University of Geneva, 24, Rue Général Dufour, 1211, Geneva 4, Switzerland

    José D. P. Rolim

  3. Department of Computer Sciences, ETH Zurich, 8092, Zurich, Switzerland

    Emo Welzl

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© 2000 Springer-Verlag Berlin Heidelberg

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Boros, E., Gurvich, V., Khachiyan, L., Makino, K. (2000). Generating Partial and Multiple Transversals of a Hypergraph. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds) Automata, Languages and Programming. ICALP 2000. Lecture Notes in Computer Science, vol 1853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45022-X_50

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  • DOI: https://doi.org/10.1007/3-540-45022-X_50

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67715-4

  • Online ISBN: 978-3-540-45022-1

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