Generating All Minimal Integral Solutions to Monotone ∧,∨-Systems of Linear, Transversal and Polymatroid Inequalities

  • L. Khachiyan
  • E. Boros
  • K. Elbassioni
  • V. Gurvich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)


We consider monotone ∨, ∧-formulae φ of m atoms, each of which is a monotone inequality of the form f i (x)≥ t i over the integers, where for i = 1,...,m, \(f_i : \mathbb{Z}^n \mapsto \mathbb{R}\) is a given monotone function and t i is a given threshold. We show that if the ∨-degree of φ is bounded by a constant, then for linear, transversal and polymatroid monotone inequalities all minimal integer vectors satisfying φ can be generated in incremental quasi-polynomial time. In contrast, the enumeration problem for the disjunction of m inequalities is NP-hard when m is part of the input. We also discuss some applications of the above results in disjunctive programming, data mining, matroid and reliability theory.


Association Rule Disjunctive Normal Form Discrete Apply Mathematic Transversal Function Monotone System 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abel, U., Bicker, R.: Determination of All Cutsets Between a Node Pair in an Undirected Graph. IEEE Transactions on Reliability 31, 167–171 (1986)CrossRefGoogle Scholar
  2. 2.
    Agrawal, R., Imielinski, T., Swami, A.: Mining association rules between sets of items in massive databases. In: Proc. the 1993 ACM-SIGMOD Int. Conf. Management of Data, pp. 207–216.Google Scholar
  3. 3.
    Agrawal, R., Mannila, H., Srikant, R., Toivonen, H., Verkamo, A.I.: Fast discovery of association rules. In: Fayyad, U.M., Piatetsky-Shapiro, G., Smyth, P., Uthurusamy, R. (eds.) Advances in Knowledge Discovery and Data Mining, pp. 307–328. AAAI Press, Menlo Park (1996)Google Scholar
  4. 4.
    Atallah, M.J., Fredrickson, G.N.: A note on finding a maximum empty rectangle. Discrete Applied Mathematics 13, 87–91 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Balas, E.: Disjunctive Programming. Annals of Discrete Mathematics 5, 3–51 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Balas, E., Zemel, E.: All the facets of zero-one programming polytopes with positive coefficients, Management Science Research Report 374, Carnegie Mellon University, Pittsburgh (1975)Google Scholar
  7. 7.
    Bioch, J.C., Ibaraki, T.: Complexity of identification and dualization of positive Boolean functions. Information and Computation 123, 50–63 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L.: Generating dual-bounded hypergraphs. Optimization Methods and Software 17, 749–781 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L., Makino, K.: Dual-bounded generating problems: All minimal integer solutions for a monotone system of linear inequalities. SIAM Journal on Computing 31(5), 1624–1643 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L.: An inequality for polymatroid functions and its applications. Discrete Applied Mathematics 131(2), 255–281 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L., Makino, K.: An Intersection Inequality for Discrete Distributions and Related Generation Problems. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 543–555. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L.: Generating all minimal integral solutions to AND-OR systems of monotone inequalities: conjunctions are easier than disjunctions, DIMACS Technical Report 2005-12, Rutgers University, New Brunswick, New Jersey, USA,
  13. 13.
    Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: Dual-bounded generating problems: partial and multiple transversals of a hypergraph. SIAM Journal on Computing 30(6), 2036–2050 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: Generating weighted transversals of a hypergraph, Dual-Bounded Generating Problems: Weighted Transversals of a Hypergraph. Discrete Applied Mathematics 142(1-3), 1–15 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Bansal, V.K., Misra, K.B., Jain, M.P.: Minimal Pathset and Minimal Cutset Using Search Technique. Microelectr. Reliability 22, 1067–1075 (1982)CrossRefGoogle Scholar
  16. 16.
    Chazelle, B. (Scot) Drysdale III, R.L., Lee, D.T.: Computing the largest empty rectangle. SIAM Journal on Computing 15(1), 550–555 (1986)CrossRefGoogle Scholar
  17. 17.
    Colburn, C.J.: The Combinatorics of Network Reliability. Oxford Univ. Press, New York (1987)Google Scholar
  18. 18.
    Edmonds, J., Gryz, J., Liang, D., Miller, R.J.: Mining for empty rectangles in large data sets. In: Van den Bussche, J., Vianu, V. (eds.) ICDT 2001. LNCS, vol. 1973, pp. 174–188. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  19. 19.
    Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing 24, 1278–1304 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Fredman, M.L., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. Journal of Algorithms 21, 618–628 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Gunopulos, D., Khardon, R., Mannila, H., Toivonen, H.: Data mining, hypergraph transversals and machine learning, in. In: Proc. the 16th ACM-SIGACT-SIGMOD-SIGART Symp. Principles of Database Systems, pp. 12–15 (1997)Google Scholar
  22. 22.
    Gurvich, V., Khachiyan, L.: On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions. Discrete Applied Mathematics 96-97, 363–373 (1999)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Lawler, E., Lenstra, J.K., Rinnooy Kan, A.H.G.: Generating all maximal independent sets: NP-hardness and polynomial-time algorithms. SIAM Journal on Computing 9, 558–565 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Liu, B., Ku, L.-P., Hsu, W.: Discovering interesting holes in data. In: Proc. 15th International Joint Conference on Artificial Intelligence, Nagoya, Japan, pp. 930–935 (1997)Google Scholar
  25. 25.
    Liu, B., Wang, K., Mun, L.-F., Qi, X.-Z.: Using decision tree induction for discovering holes in data. In: Proc. 5th Pacific Rim International Conference on Artificial Intelligence, pp. 182–193 (1998)Google Scholar
  26. 26.
    Pfetsch, M.: The Maximum feasible Subsystem Problem and Vertex-Facet Incidences of Polyhedra, Dissertation, TU Berlin (2002)Google Scholar
  27. 27.
    Provan, J.S., Ball, M.O.: Computing Network Reliability in Time Polynomial in the Number of Cuts. Operations Research 32, 516–526 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Stork, F., Uetz, M.: On the Generation of Circuits and Minimal Forbidden Sets. Mathematical Programming, Series A 102(1), 185–203 (2005)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • L. Khachiyan
    • 1
  • E. Boros
    • 2
  • K. Elbassioni
    • 3
  • V. Gurvich
    • 2
  1. 1.Department of Computer ScienceRutgers UniversityPiscatawayUSA
  2. 2.RUTCORRutgers UniversityPiscatawayUSA
  3. 3.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations