Advertisement

k-Gap Interval Graphs

  • Fedor V. Fomin
  • Serge Gaspers
  • Petr Golovach
  • Karol Suchan
  • Stefan Szeider
  • Erik Jan van Leeuwen
  • Martin Vatshelle
  • Yngve Villanger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)

Abstract

We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a k-gap interval graph if it has a multiple interval representation with at most n + k intervals in total. In order to scale up the nice algorithmic properties of interval graphs (where k = 0), we parameterize graph problems by k, and find FPT algorithms for several problems, including Feedback Vertex Set, Dominating Set, Independent Set, Clique, Clique Cover, and Multiple Interval Transversal. The Coloring problem turns out to be Open image in new window -hard and we design an XP algorithm for the recognition problem.

Keywords

Maximal Clique Interval Graph Tree Decomposition Interval Number Graph Class 
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.
    Alon, N.: Piercing d-intervals. Discret. Comput. Geom. 19(3), 333–334 (1998)zbMATHCrossRefGoogle Scholar
  2. 2.
    Andreae, T.: On an extremal problem concerning the interval number of a graph. Discrete Appl. Math. 14(1), 1–9 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Andreae, T., Aigner, M.: The total interval number of a graph. J. Comb. Theory Ser. B 46(1), 7–21 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Aumann, Y., Lewenstein, M., Melamud, O., Pinter, R.Y., Yakhini, Z.: Dotted interval graphs and high throughput genotyping. In: SODA 2005, pp. 339–348 (2005)Google Scholar
  5. 5.
    Bafna, V., Narayanan, B.O., Ravi, R.: Nonoverlapping local alignments (weighted independent sets of axis-parallel rectangles). Discrete Appl. Math. 71(1-3), 41–53 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Balogh, J., Ochem, P., Pluhár, A.: On the interval number of special graphs. J. Graph Theor. 46(4), 241–253 (2004)zbMATHCrossRefGoogle Scholar
  7. 7.
    Bar-Yehuda, R., Halldórsson, M.M., Naor, J., Shachnai, H., Shapira, I.: Scheduling split intervals. SIAM J. Comput. 36(1), 1–15 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bar-Yehuda, R., Rawitz, D.: Using fractional primal-dual to schedule split intervals with demands. Discrete Optim. 3(4), 275–287 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Belmonte, R., Vatshelle, M.: Graph Classes with Structured Neighborhoods and Algorithmic Applications. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 47–58. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Bessière, C., Hebrard, E., Hnich, B., Kiziltan, Z., Quimper, C.-G., Walsh, T.: The parameterized complexity of global constraints. In: AAAI 2008, pp. 235–240 (2008)Google Scholar
  11. 11.
    Blin, G., Fertin, G., Vialette, S.: Extracting constrained 2-interval subsets in 2-interval sets. Theor. Comput. Sci. 385(1-3), 241–263 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209(1-2), 1–45 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms. J. Comput. System Sci. 13(3), 335–379 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bui-Xuan, B.-M., Telle, J.A., Vatshelle, M.: Boolean-width of graphs. Theor. Comput. Sci. 412(39), 5187–5204 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Butman, A., Hermelin, D., Lewenstein, M., Rawitz, D.: Optimization problems in multiple-interval graphs. ACM Trans. Algorithms 6(2) (2010)Google Scholar
  16. 16.
    Catlin, P.A.: Supereulerian graphs: A survey. J. Graph Theor. 16(2), 177–196 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Chen, E., Yang, L., Yuan, H.: Improved algorithms for largest cardinality 2-interval pattern problem. J. Comb. Optim. 13(3), 263–275 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Chen, M., Chang, G.J.: Total interval numbers of complete r-partite graphs. Discrete Appl. Math. 122, 83–92 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Courcelle, B.: The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues. Rairo - Theor. Inform. Appl. 26, 257–286 (1992)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Crochemore, M., Hermelin, D., Landau, G.M., Rawitz, D., Vialette, S.: Approximating the 2-interval pattern problem. Theor. Comput. Sci. 395(2-3), 283–297 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer (1999)Google Scholar
  22. 22.
    Erdös, P., West, D.B.: A note on the interval number of a graph. Discrete Math. 55(2), 129–133 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Fellows, M.R., Hermelin, D., Rosamond, F., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410, 53–61 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series XIV. Springer (2006)Google Scholar
  25. 25.
    Fomin, F.V., Gaspers, S., Golovach, P., Suchan, K., Szeider, S., van Leeuwen, E.J., Vatshelle, M., Villanger, Y.: k-gap interval graphs. arXiv CoRR 1112.3244 (2011)Google Scholar
  26. 26.
    Gambette, P., Vialette, S.: On Restrictions of Balanced 2-Interval Graphs. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 55–65. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  27. 27.
    Gaspers, S., Szeider, S.: Kernels for global constraints. In: IJCAI 2011, pp. 540–545 (2011)Google Scholar
  28. 28.
    Golumbic, M.C.: Algorithmic graph theory and perfect graphs. Academic Press (1980)Google Scholar
  29. 29.
    Griggs, J.R., West, D.B.: Extremal values of the interval number of a graph. SIAM J. Algebra. Discr. 1(1), 1–7 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Hassin, R., Segev, D.: Rounding to an integral program. Oper. Res. Lett. 36(3), 321–326 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Jansen, B.M.P., Kratsch, S.: Data Reduction for Graph Coloring Problems. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 90–101. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  32. 32.
    Jiang, M., Zhang, Y.: Parameterized Complexity in Multiple-interval Graphs: Domination. In: Rossmanith, P. (ed.) IPEC 2011. LNCS, vol. 7112, pp. 27–40. Springer, Heidelberg (2012)Google Scholar
  33. 33.
    Jiang, M., Zhang, Y.: Parameterized Complexity in Multiple-Interval Graphs: Partition, Separation, Irredundancy. In: Fu, B., Du, D.-Z. (eds.) COCOON 2011. LNCS, vol. 6842, pp. 62–73. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  34. 34.
    Kaiser, T.: Transversals of d-intervals. Discret. Comput. Geom. 18(2) (1997)Google Scholar
  35. 35.
    Kloks, T.: Treewidth, Computations and Approximations. LNCS, vol. 842. Springer, Heidelberg (1994)zbMATHGoogle Scholar
  36. 36.
    Kostochka, A.V., West, D.B.: Total interval number for graphs with bounded degree. J. Graph Theor. 25(1), 79–84 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Kratzke, T.M., West, D.B.: The total interval number of a graph, I: Fundamental classes. Discrete Math. 118(1-3), 145–156 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Kratzke, T.M., West, D.B.: The total interval number of a graph II: Trees and complexity. SIAM J. Discrete Math. 9(2), 339–348 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Marx, D.: Parameterized coloring problems on chordal graphs. Theor. Comput. Sci. 351(3), 407–424 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Marx, D.: Chordal deletion is fixed-parameter tractable. Algorithmica 57(4), 747–768 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press (2006)Google Scholar
  42. 42.
    Raychaudhuri, A.: The total interval number of a tree and the hamiltonian completion number of its line graph. Inform. Process. Lett. 56(6), 299–306 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Scheinerman, E.R., West, D.B.: The interval number of a planar graph: Three intervals suffice. J. Comb. Theory Ser. B 35(3), 224–239 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Spinrad, J.P.: Efficient Graph Representations. Fields Institute Monographs, vol. 19. AMS (2003)Google Scholar
  46. 46.
    Tardos, G.: Transversals of 2-intervals, a topological approach. Combinatorica 15(1), 123–134 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Trotter, W.T., Harary, F.: On double and multiple interval graphs. J. Graph Theor. 3(3), 205–2011 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. SIAM J. Comput. 6(3), 505–517 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Vialette, S.: On the computational complexity of 2-interval pattern matching problems. Theor. Comput. Sci. 312(2-3), 224–239 (2004)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Vialette, S.: Two-interval pattern problems. In: Encyclopedia of Algorithms. Springer (2008)Google Scholar
  51. 51.
    West, D.B.: A short proof of the degree bound for interval number. Discrete Math. 73(3), 309–310 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    West, D.B., Shmoys, D.B.: Recognizing graphs with fixed interval number is NP-complete. Discrete Appl. Math. 8, 295–305 (1984)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Fedor V. Fomin
    • 1
  • Serge Gaspers
    • 2
  • Petr Golovach
    • 3
  • Karol Suchan
    • 4
    • 5
  • Stefan Szeider
    • 2
  • Erik Jan van Leeuwen
    • 1
  • Martin Vatshelle
    • 1
  • Yngve Villanger
    • 1
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.Inst. of Information SystemsVienna University of TechnologyViennaAustria
  3. 3.School of Engineering and Computing SciencesDurham UniversityDurhamUK
  4. 4.Facultad de Ingeniería y CienciasUniversidad Adolfo IbáñezSantiagoChile
  5. 5.Faculty of Applied Mathematics WMSAGH - University of Science and TechnologyKrakowPoland

Personalised recommendations