Abstract
The problem of determining the size of a finite concept lattice is shown to be #P-complete. Since any finite lattice can be represented as a concept lattice, the problem of determining the size of a lattice given by the ordered sets of its irreducibles is also #P-complete. Some results about NP-completeness or polynomial tractability of decision problems related to concepts with bounded extent, intent, and the sum of both are given. These problems can be reformulated as decision problems about lattice elements generated by a certain amount of irreducibles.
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References
Barbut, M. and Monjardet, B. (1970) Ordre et Classification, Algèbre et Combinatoire, 2 tomes, Hachette, Paris.
Bordat, J. P. (1986) Calcul pratique du treillis de Galois d'une correspondance,Math. Sci. Hum. 96, 31–47.
Ganter, B. (1984) Two basic algorithms in concept analysis, Preprint 831, Technische Hochschule Darmstadt.
Ganter, B. and Kuznetsov, S. (2000) Formalizing hypotheses with concepts, in G. Mineau and B. Ganter (eds), Proc. of the 8th Internat. Conf. on Conceptual Structures, ICCS'2000, Lecture Notes in Artificial Intelligence 1867, pp. 342–356.
Ganter, B. and Wille, R. (1999) Formal Concept Analysis: Mathematical Foundations, Springer, Berlin.
Garey, M. and Johnson, D. (1979) Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, New York.
Habib, M. and Nourine, L. (1996) Tree structure for distributive lattices and its applications, Theoret. Comput. Sci. 165(2), 391–405.
Harary, F. (1969) Graph Theory, Addision-Wesley, Reading, MA.
Karp, R. and Hopcroft, G. A. (1973) 2.5-algorithm for maximum matching in bipartite graphs, SIAM J. Comput. 2(2), 225–231.
Kuznetsov, S. O. and Objedkov, S. A. (2001) Comparing performance of algorithms for generating concept lattices, in E. Mephu Nguifo et al. (eds), Proc. of the ICCS-2001 International Workshop on Concept Lattices-Based Theory, Methods and Tools for Knowledge Discovery in Databases (CLKDD '01), Stanford University, Palo Alto, pp. 35–47.
Norris, E. M. (1978) An algorithm for computing the maximal rectangles in a binary relation, Rev. Roumaine Math. Pures Appl. 23(2), 243–250.
Nourine, L. and Raynaud, O. (1999) A fast algorithm for building lattices, Inform. Process. Lett. 71, 199–204.
Schütt, D. (1988) Abschôtzungen für die Anzahl der Begriffe von Kontexten, Diplomarbeit TH Darmstadt, Darmstadt.
Valiant, L. G. (1979) The complexity of enumeration and reliability problems, SIAMJ. Comput. 8(3), 410–421.
Wille, R. (1982) Restructuring lattice theory: An approach based on hierarchies of concepts, in I. Rival (ed.), Ordered Sets, Reidel, Dordrecht/Boston, pp. 445–470.
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Kuznetsov, S.O. On Computing the Size of a Lattice and Related Decision Problems. Order 18, 313–321 (2001). https://doi.org/10.1023/A:1013970520933
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DOI: https://doi.org/10.1023/A:1013970520933