Advertisement

Algorithms Parameterized by Vertex Cover and Modular Width, through Potential Maximal Cliques

  • Fedor V. Fomin
  • Mathieu Liedloff
  • Pedro Montealegre
  • Ioan Todinca
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8503)

Abstract

In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover (vc) and modular width (mw). We prove that for any graph, the number of minimal separators is \(\mathcal{O}^*(3^{\operatorname{vc}})\) and \(\mathcal{O}^*(1.6181^{\operatorname{mw}})\), the number of potential maximal cliques is \(\mathcal{O}^*(4^{\operatorname{vc}})\) and \(\mathcal{O}^*(1.7347^{\operatorname{mw}})\), and these objects can be listed within the same running times. (The \(\mathcal{O}^*\) notation suppresses polynomial factors in the size of the input.) Combined with known results [3,12], we deduce that a large family of problems, e.g., Treewidth, Minimum Fill-in, Longest Induced Path, Feedback vertex set and many others, can be solved in time \(\mathcal{O}^*(4^{\operatorname{vc}})\) or \(\mathcal{O}^*(1.7347^{\operatorname{mw}})\).

Keywords

Vertex Cover Maximal Clique Input Graph Minimal Separator Minimum Vertex Cover 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bodlaender, H.L., Fomin, F.V.: Tree decompositions with small cost. Discrete Applied Mathematics 145(2), 143–154 (2005)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bodlaender, H.L., Rotics, U.: Computing the treewidth and the minimum fill-in with the modular decomposition. Algorithmica 36(4), 375–408 (2003)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bouchitté, V., Todinca, I.: Treewidth and minimum fill-in: Grouping the minimal separators. SIAM J. Comput. 31(1), 212–232 (2001)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bouchitté, V., Todinca, I.: Listing all potential maximal cliques of a graph. Theor. Comput. Sci. 276(1-2), 17–32 (2002)CrossRefMATHGoogle Scholar
  5. 5.
    Chapelle, M., Liedloff, M., Todinca, I., Villanger, Y.: Treewidth and pathwidth parameterized by the vertex cover number. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 232–243. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  6. 6.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    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)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Cygan, M., Lokshtanov, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: On cutwidth parameterized by vertex cover. Algorithmica 68(4), 940–953 (2014)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Fellows, M.R., Lokshtanov, D., Misra, N., Rosamond, F.A., Saurabh, S.: Graph layout problems parameterized by vertex cover. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 294–305. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Fomin, F.V., Kratsch, D., Todinca, I., Villanger, Y.: Exact algorithms for treewidth and minimum fill-in. SIAM J. Comput. 38(3), 1058–1079 (2008)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Fomin, F.V., Liedloff, M., Montealegre, P., Todinca, I.: Algorithms parameterized by vertex cover and modular width, through potential maximal cliques (2014), http://arxiv.org/abs/1404.3882
  12. 12.
    Fomin, F.V., Todinca, I., Villanger, Y.: Large induced subgraphs via triangulations and cmso. In: Chekuri, C. (ed.) SODA, pp. 582–583. SIAM (2014), http://arxiv.org/abs/1309.1559
  13. 13.
    Fomin, F.V., Villanger, Y.: Finding induced subgraphs via minimal triangulations. In: Marion, J.Y., Schwentick, T. (eds.) STACS. LIPIcs, vol. 5, pp. 383–394. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2010)Google Scholar
  14. 14.
    Fomin, F.V., Villanger, Y.: Treewidth computation and extremal combinatorics. Combinatorica 32(3), 289–308 (2012)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. Ann. Pure Appl. Logic 130(1-3), 3–31 (2004)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Gajarský, J., Lampis, M., Ordyniak, S.: Parameterized algorithms for modular-width. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 163–176. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  17. 17.
    Gysel, R.: Potential maximal clique algorithms for perfect phylogeny problems. CoRR, abs/1303.3931 (2013)Google Scholar
  18. 18.
    Lampis, M.: Algorithmic meta-theorems for restrictions of treewidth. Algorithmica 64(1), 19–37 (2012)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Lokshtanov, D.: On the complexity of computing treelength. Discrete Applied Mathematics 158(7), 820–827 (2010)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Tedder, M., Corneil, D.G., Habib, M., Paul, C.: Simpler linear-time modular decomposition via recursive factorizing permutations. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 634–645. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Fedor V. Fomin
    • 1
  • Mathieu Liedloff
    • 2
  • Pedro Montealegre
    • 2
  • Ioan Todinca
    • 2
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.INSA Centre Val de Loire, LIFO EA 4022Univ. OrléansOrléans Cedex 2France

Personalised recommendations