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Decomposition Width of Matroids

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Automata, Languages and Programming (ICALP 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6198))

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Abstract

Hliněný [J. Combin. Theory Ser. B 96 (2006), 325–351] showed that every matroid property expressible in the monadic second order logic can be decided in linear time for matroids with bounded branch-width that are represented over finite fields. To be able to extend these algorithmic results to matroids not representable over finite fields, we introduce a new matroid width parameter, the decomposition width, and show that every matroid property expressible in the monadic second order logic can be computed in linear time for matroids given by a decomposition with bounded width. We also relate the decomposition width to matroid branch-width and discuss implications of our results with respect to other known algorithms.

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References

  1. Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of fiding embeddings in a k-tree. SIAM J. Alg. Disc. Meth. 8, 277–284 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree decomposable graphs. J. Algorithms 12, 308–340 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bodlaender, H.: Dynamic programming algorithms on graphs with bounded tree-width. In: Lepistö, T., Salomaa, A. (eds.) ICALP 1988. LNCS, vol. 317, pp. 105–119. Springer, Heidelberg (1988)

    Google Scholar 

  4. Bodlaender, H.: A linear time algorithm for finding tree-decompositions of small treewidth. In: Proc. SODA 1993, pp. 226–234. ACM & SIAM (1993)

    Google Scholar 

  5. Courcelle, B.: The monadic second-order logic of graph I. Recognizable sets of finite graphs. Inform. and Comput. 85, 12–75 (1990)

    MATH  MathSciNet  Google Scholar 

  6. Courcelle, B.: The expression of graph properties and graph transformations in monadic second-order logic. In: Rozenberg, G. (ed.) Handbook of graph grammars and computing by graph transformations. Foundations, vol. 1, pp. 313–400. World Scientific, Singapore (1997)

    Google Scholar 

  7. Geelen, J., Gerards, B., Whittle, G.: Branch-width and well-quasi-ordering in matroids and graphs. J. Combin. Theory Ser. B 84, 270–290 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Geelen, J., Gerards, B., Whittle, G.: On Rota’s Conjecture and excluded minors containing large projective geometries. J. Combin. Theory Ser. B 96, 405–425 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Geelen, J., Gerards, B., Whittle, G.: Excluding a planar graph from GF(q)-representable matroids. J. Combin. Theory Ser. B 97, 971–998 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Geelen, J., Gerards, B., Whittle, G.: Tangles, tree-decompositions, and grids in matroids. J. Combin. Theory Ser. B 97, 657–667 (2009)

    Article  MathSciNet  Google Scholar 

  11. Hliněný, P.: Branch-width, parse trees, and monadic second-order logic for matroids. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 319–330. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  12. Hliněný, P.: On matroid properties definable in the MSO logic. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 470–479. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  13. Hliněný, P.: A parametrized algorithm for matroid branch-width. SIAM J. Computing 35, 259–277 (2005)

    Article  MATH  Google Scholar 

  14. Hliněný, P.: Branch-width, parse trees, and monadic second-order logic for matroids, J. Combin. Theory Ser. B 96, 325–351 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hliněný, P.: On matroid representability and minor problems. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 505–516. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  16. Hliněný, P.: The Tutte polynomial for matroids of bounded branch-width. Combin. Probab. Comput. 15, 397–406 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hliněný, P., Oum, S.: Finding branch-decomposition and rank-decomposition. SIAM J. Computing 38, 1012–1032 (2008)

    Article  MATH  Google Scholar 

  18. Hliněný, P., Whittle, G.: Matroid tree-width. Europ. J. Combin. 27, 1117–1128 (2006)

    Article  MATH  Google Scholar 

  19. Hliněný, P., Whittle, G.: Addendum to Matroid tree-Width. Europ. J. Combin. 30, 1036–1044 (2009)

    Article  MATH  Google Scholar 

  20. Král’, D.: Computing representations of matroids of bounded branch-width. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 224–235. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  21. Oum, S., Seymour, P.D.: Approximating clique-width and branch-width. J. Combin. Theory Ser. B 96, 514–528 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Oum, S., Seymour, P.D.: Testing branch-width. J. Combin. Theory Ser. B 97, 385–393 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Oxley, J.G.: Matroid theory. Oxford Graduate Texts in Mathematics, vol. 3. Oxford University Press, Oxford (1992)

    MATH  Google Scholar 

  24. Seymour, P.: Recognizing graphic matroids. Combinatorica 1, 75–78 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  25. Strozecki, Y.: A logical approach to decomposable matroids, arXiv 0908.4499

    Google Scholar 

  26. Truemper, K.: Matroid decomposition. Academic Press, London (1992)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Theoretical Computer Science (ITI) Faculty of Mathematics and Physics, Charles University, Malostranské náměstí 25, Prague, Czech Republic

    Daniel Král’

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  1. Daniel Král’
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Editor information

Editors and Affiliations

  1. Oxford University Computing Laboratory, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Samson Abramsky

  2. Université de Bordeaux (LaBRI) & INRIA, 351, cours de la Libération, 33405, Talence Cedex, France

    Cyril Gavoille

  3. INRIA, Centre de Recherche Bordeaux – Sud-Ouest, 351 cours de la Libération, 33405, Talence Cedex, France

    Claude Kirchner

  4. Heinz Nixdorf Institute, University of Paderborn, Fürstenallee 11, 33102, Paderborn, Germany

    Friedhelm Meyer auf der Heide

  5. University of Patras and RACTI, 26500, Patras, Greece

    Paul G. Spirakis

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Král’, D. (2010). Decomposition Width of Matroids. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds) Automata, Languages and Programming. ICALP 2010. Lecture Notes in Computer Science, vol 6198. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14165-2_6

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  • DOI: https://doi.org/10.1007/978-3-642-14165-2_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14164-5

  • Online ISBN: 978-3-642-14165-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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