Solving Problems on Graphs of High Rank-Width

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)

Abstract

A modulator of a graph G to a specified graph class \({\mathcal H}\) is a set of vertices whose deletion puts G into \({\mathcal H}\). The cardinality of a modulator to various graph classes has long been used as a structural parameter which can be exploited to obtain FPT algorithms for a range of hard problems. Here we investigate what happens when a graph contains a modulator which is large but “well-structured” (in the sense of having bounded rank-width). Can such modulators still be exploited to obtain efficient algorithms? And is it even possible to find such modulators efficiently?

We first show that the parameters derived from such well-structured modulators are strictly more general than the cardinality of modulators and rank-width itself. Then, we develop an FPT algorithm for finding such well-structured modulators to any graph class which can be characterized by a finite set of forbidden induced subgraphs. We proceed by showing how well-structured modulators can be used to obtain efficient parameterized algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use the concept of well-structured modulators to develop an algorithmic meta-theorem for efficiently deciding problems expressible in Monadic Second Order (MSO) logic, and prove that this result is tight in the sense that it cannot be generalized to LinEMSO problems.

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References

  1. 1.
    Alekseev, V.E.: Polynomial algorithm for finding the largest independent sets in graphs without forks. Discr. Appl. Math. 135(1–3), 3–16 (2004)Google Scholar
  2. 2.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Kernel bounds for path and cycle problems. Theor. Comput. Sci. 511, 117–136 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brandstädt, A., Lozin, V.V.: A note on alpha-redundant vertices in graphs. Discr. Appl. Math. 108(3), 301–308 (2001)CrossRefMATHGoogle Scholar
  4. 4.
    Cai, L.: Parameterized complexity of vertex colouring. Discr. Appl. Math. 127(3), 415–429 (2003)CrossRefMATHGoogle Scholar
  5. 5.
    Corneil, D.G., Lerchs, H., Burlingham, L.S.: Complement reducible graphs. Discr. Appl. Math. 3, 163–174 (1981)MathSciNetMATHGoogle Scholar
  6. 6.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cunningham, W.H.: Decomposition of directed graphs. SIAM J. Algebraic Discrete Methods 3(2), 214–228 (1982)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Diestel, R.: Graph Theory. GTM, vol. 173, 2nd edn. Springer Verlag, New York (2000)Google Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer Verlag (2013)Google Scholar
  10. 10.
    Gajarský, J., Hliněný, P., Obdržálek, J., Ordyniak, S., Reidl, F., Rossmanith, P., Villaamil, F.S., Sikdar, S.: Kernelization using structural parameters on sparse graph classes. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 529–540. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  11. 11.
    Ganian, R., Hliněný, P.: On parse trees and Myhill-Nerode-type tools for handling graphs of bounded rank-width. Discr. Appl. Math. 158(7), 851–867 (2010)CrossRefMATHGoogle Scholar
  12. 12.
    Gaspers, S., Misra, N., Ordyniak, S., Szeider, S., Živný, S.: Backdoors into heterogeneous classes of SAT and CSP. In: Brodley, C.E., Stone, P.(eds.), Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, pp. 2652–2658. AAAI Press (2014)Google Scholar
  13. 13.
    Gerber, M.U., Lozin, V.V.: Robust algorithms for the stable set problem. Graphs and Combinatorics 19(3), 347–356 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gioan, E., Paul, C.: Dynamic distance hereditary graphs using split decomposition. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 41–51. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  15. 15.
    Gioan, E., Paul, C.: Split decomposition and graph-labelled trees: characterizations and fully dynamic algorithms for totally decomposable graphs. Discr. Appl. Math. 160(6), 708–733 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gioan, E., Paul, C., Tedder, M., Corneil, D.: Practical and efficient split decomposition via graph-labelled trees. Algorithmica 69(4), 789–843 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Golovach, P.A., Paulusma, D., Song, J.: Closing complexity gaps for coloring problems on h-free graphs. Inf. Comput. 237, 204–214 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hliněný, P., Oum, S.I.: Finding branch-decompositions and rank-decompositions. SIAM J. Comput. 38(3), 1012–1032 (2008)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kochol, M., Lozin, V.V., Randerath, B.: The 3-colorability problem on graphs with maximum degree four. SIAM J. Comput. 32(5), 1128–1139 (2003)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Libkin, L.: Elements of Finite Model Theory. Springer (2004)Google Scholar
  21. 21.
    Lokshantov, D., Vatshelle, M., Villanger, Y.: Independent set in p\(_{\text{5 }}\)-free graphs in polynomial time. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pp. 570–581. SIAM (2014)Google Scholar
  22. 22.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2006) CrossRefMATHGoogle Scholar
  23. 23.
    Oum, S., Seymour, P.: Approximating clique-width and branch-width. J. Combin. Theory Ser. B 96(4), 514–528 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Algorithms and Complexity Group, Institute of Computer Graphics and AlgorithmsTU WienViennaAustria

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