Largest Chordal and Interval Subgraphs Faster Than 2n

  • Ivan Bliznets
  • Fedor V. Fomin
  • Michał Pilipczuk
  • Yngve Villanger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8125)


We prove that in an n-vertex graph, induced chordal and interval subgraphs with the maximum number of vertices can be found in time \(\mathcal{O}(2^{\lambda n})\) for some λ < 1. These are the first algorithms breaking the trivial 2 n n O(1) bound of the brute-force search for these problems.


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  1. 1.
    Brandstädt, A., Le, V., Spinrad, J.P.: Graph Classes. A Survey, SIAM Mon. on Discrete Mathematics and Applications. SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  2. 2.
    Fomin, F.V., Gaspers, S., Kratsch, D., Liedloff, M., Saurabh, S.: Iterative compression and exact algorithms. Theor. Comput. Sci. 411, 1045–1053 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Fomin, F.V., Gaspers, S., Pyatkin, A.V., Razgon, I.: On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Springer (2010)Google Scholar
  5. 5.
    Fomin, F.V., Todinca, I., Villanger, Y.: Exact algorithm for the maximum induced planar subgraph problem. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 287–298. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    Fomin, F.V., Villanger, Y.: Treewidth computation and extremal combinatorics. Combinatorica 32, 289–308 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gaspers, S., Kratsch, D., Liedloff, M.: On independent sets and bicliques in graphs. Algorithmica 62, 637–658 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)zbMATHGoogle Scholar
  9. 9.
    Gupta, S., Raman, V., Saurabh, S.: Maximum r-regular induced subgraph problem: Fast exponential algorithms and combinatorial bounds. SIAM J. Discrete Math. 26, 1758–1780 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Lekkerkerker, C.G., Boland, J.C.: Representation of a finite graph by a set of intervals on the real line. Fund. Math. 51, 45–64 (1962)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20, 219–230 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Pilipczuk, M., Pilipczuk, M.: Finding a maximum induced degenerate subgraph faster than 2n. In: Thilikos, D.M., Woeginger, G.J. (eds.) IPEC 2012. LNCS, vol. 7535, pp. 3–12. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  13. 13.
    Raman, V., Saurabh, S., Sikdar, S.: Efficient exact algorithms through enumerating maximal independent sets and other techniques. Theory Comput. Syst. 41, 563–587 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Robson, J.M.: Algorithms for maximum independent sets. J. Algorithms 7, 425–440 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Schroeppel, R., Shamir, A.: A T = O(2n/2), S = O(2n/4) algorithm for certain NP-complete problems. SIAM J. Comput. 10, 456–464 (1981)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ivan Bliznets
    • 1
  • Fedor V. Fomin
    • 2
  • Michał Pilipczuk
    • 2
  • Yngve Villanger
    • 2
  1. 1.St. Petersburg Academic University of the Russian Academy of SciencesRussia
  2. 2.Department of InformaticsUniversity of BergenNorway

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