Combinatorial Problems With Closure Structures

Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 110)


We consider a specific class of combinatorial search respectively optimization problems where the search space gives rise to a closure operator and essentially the hulls are the only relevant subsets that must be checked in a brute force approach. We suggest that such a closure structure can help to reduce time complexities. Moreover we propose two types of (structural) parameterizations of instance classes based on the closure property and outline how it could be used to achieve fixed-parameter tractability (FPT) characterizations. In this setting, three example problems are described: a covering problem from combinatorial geometry, a variant of the autarky problem in propositional logic, and a specific graph problem on finite forests.


Exact algorithmics Closure operator FPT Combinatorial optimization Computational complexity 


  1. 1.
    Boissonnat JD, Yvinec M (1998) Algorithmic geometry, Cambridge University Press, CambridgeMATHGoogle Scholar
  2. 2.
    Chen J, Kanj IA, Xia G (2010) Improved upper bounds for vertex cover. Theor Comp Sci 411:3736–3756CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Cook SA (1971) The complexity of theorem proving procedures. In: Proceedings of the 3rd ACM Symposium on Theory of Computing, ACM, Ohio, USA, pp 151–158Google Scholar
  4. 4.
    Downey RG, Fellows MR (1999) Parameterized complexity. Springer, New YorkCrossRefGoogle Scholar
  5. 5.
    Franco J, Goldsmith J, Schlipf J, Speckenmeyer E, Swaminathan RP (1999) An algorithm for the class of pure implicational formulas. Discrete Appl Math 96:89–106CrossRefMathSciNetGoogle Scholar
  6. 6.
    Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman and Company, San FranciscoMATHGoogle Scholar
  7. 7.
    Golumbic MC (1980) Algorithmic graph theory and perfect graphs. Academic Press, New YorkMATHGoogle Scholar
  8. 8.
    Monien B, Speckenmeyer E (1985) Solving satisfiability in less than 2n steps. Discrete Appl Math 10:287–295CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Porschen S (2005) On the rectangular subset closure of point sets. Proc ICCSA/CGA 2005, LNCS 3480:796–805Google Scholar
  10. 10.
    Porschen S (2006) Algorithms for rectangular covering problems. Proc ICCSA/CGA 2006, LNCS 3980:40–49Google Scholar
  11. 11.
    Porschen S (2007) A CNF formula hierarchy over the hypercube. Proc AI 2007, LNAI 4830:234–243MathSciNetGoogle Scholar
  12. 12.
    Porschen S (2009) An FPT-variant of the shadow problem with Kernelization. Proc ICCS 2009, 432–439, Hong KongGoogle Scholar
  13. 13.
    Porschen S (2011) On problems with closure properties. Lecture notes in engineering and computer science: Proc IMECS 2011, pp. 258–262, 16–18 March, 2011, Hong KongGoogle Scholar
  14. 14.
    Birkhoff G (1995) Lattice theory. American Mathematical Society, Providence, Rhode IslandGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Mathematics Group, Department 4HTW BerlinBerlinGermany

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