Fixed-Parameter Algorithms for Cochromatic Number and Disjoint Rectangle Stabbing

  • Pinar Heggernes
  • Dieter Kratsch
  • Daniel Lokshtanov
  • Venkatesh Raman
  • Saket Saurabh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6139)


Given a permutation π of {1,...,n} and a positive integer k, we give an algorithm with running time \(2^{O(k^2 \log k)}n^{O(1)}\) that decides whether π can be partitioned into at most k increasing or decreasing subsequences. Thus we resolve affirmatively the open question of whether the problem is fixed parameter tractable. This NP-complete problem is equivalent to deciding whether the cochromatic number (the minimum number of cliques and independent sets the vertices of the graph can be partitioned into) of a given permutation graph on n vertices is at most k. In fact, we give a more general result: within the mentioned running time, one can decide whether the cochromatic number of a given perfect graph on n vertices is at most k.

To obtain our result we use a combination of two well-known techniques within parameterized algorithms, namely greedy localization and iterative compression. We further demonstrate the power of this combination by giving a \(2^{O(k^2 \log k)}n \log n\) time algorithm for deciding whether a given set of n non-overlapping axis-parallel rectangles can be stabbed by at most k of the given set of horizontal and vertical lines. Whether such an algorithm exists was mentioned as an open question in several papers.


Chromatic Number Maximum Clique Recursive Call Perfect Graph Clique Number 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berge, C.: Färbung von Graphen, deren sämtliche bzw. deren ungeraden Kreise starr sind (Zusammenfassung), Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur., Reihe 10,  114 (1961)Google Scholar
  2. 2.
    Brandstädt, A.: Partitions of graphs into one or two independent sets and cliques. Discrete Mathematics 152, 47–54 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brandstädt, A., Kratsch, D.: On the partition of permutations into increasing or decreasing subsequences. Elektron. Inform. Kybernet. 22, 263–273 (1986)zbMATHGoogle Scholar
  4. 4.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM, Philadelphia (1999)zbMATHGoogle Scholar
  5. 5.
    Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55 (2008)Google Scholar
  6. 6.
    Chen, J., Fomin, F.V., Liu, Y., Lu, S., Villanger, Y.: Improved algorithms for feedback vertex set problems. J. Comput. Syst. Sci. 74, 1188–1198 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chudnovsky, M., Cornuejols, G., Liu, X., Seymour, P., Vuskovic, K.: Recognizing berge graphs. Combinatorica 25, 143–186 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gaur, T.I.D.R., Krishnamurti, R.: Constant ratio approximation algorithms for the rectangle stabbing problem and the rectilinear partitioning problem. Journal of Algorithms 43, 138–152 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dehne, F.K.H.A., Fellows, M.R., Rosamond, F.A., Shaw, P.: Greedy localization, iterative compression, modeled crown reductions: New FPT techniques, an improved algorithm for set splitting, and a novel 2k kernelization for vertex cover. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 271–280. Springer, Heidelberg (2004)Google Scholar
  10. 10.
    Dom, M., Fellows, M.R., Rosamond, F.A.: Parameterized complexity of stabbing rectangles and squares in the plane. In: Das, S., Uehara, R. (eds.) WALCOM 2009. LNCS, vol. 5431, pp. 298–309. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Dom, M., Sikdar, S.: The parameterized complexity of the rectangle stabbing problem and its variants. In: Preparata, F.P., Wu, X., Yin, J. (eds.) FAW 2008. LNCS, vol. 5059, pp. 288–299. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Downey, R.D., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  13. 13.
    Erdős, P., Gimbel, J.: Some problems and results in cochromatic theory. In: Quo Vadis, Graph Theory?, pp. 261–264. North-Holland, Amsterdam (1993)Google Scholar
  14. 14.
    Erdős, P., Gimbel, J., Kratsch, D.: Extremal results in cochromatic and dichromatic theory. Journal of Graph Theory 15, 579–585 (1991)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Mathematica 2, 463–470 (1935)MathSciNetGoogle Scholar
  16. 16.
    Fishburn, P.C.: Interval Orders and Interval Graphs: A Study of Partially Ordered Sets. Wiley, Chichester (1985)zbMATHGoogle Scholar
  17. 17.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)Google Scholar
  18. 18.
    Fomin, F.V., Iwama, K., Kratsch, D., Kaski, P., Koivisto, M., Kowalik, L., Okamoto, Y., van Rooij, J., Williams, R.: 08431 Open problems – moderately exponential time algorithms. In: Fomin, F.V., Iwama, K., Kratsch, D. (eds.) Moderately Exponential Time Algorithms, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany. Dagstuhl Seminar Proceedings, vol. 08431 (2008)Google Scholar
  19. 19.
    Fomin, F.V., Kratsch, D., Novelli, J.-C.: Approximating minimum cocolorings. Inf. Process. Lett. 84, 285–290 (2002)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Frank, A.: On chain and antichain families of a partially ordered set. J. Comb. Theory, Ser. B 29, 176–184 (1980)zbMATHCrossRefGoogle Scholar
  21. 21.
    Giannopoulos, P., Knauer, C., Rote, G., Werner, D.: Fixed-parameter tractability and lower bounds for stabbing problems. In: Proceedings of the 25th European Workshop on Computational Geometry (EuroCG), pp. 281–284 (2009)Google Scholar
  22. 22.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, 2nd edn., vol. 57. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
  23. 23.
    Grötschel, M., Lovász, L., Schrijver, A.: Polynomial algorithms for perfect graph. Annals of Discrete Mathematics 21, 325–356 (1984)Google Scholar
  24. 24.
    Hassin, R., Megiddo, N.: Approximation algorithms for hitting objects with straight lines. Discrete Applied Mathematics 30, 29–42 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Jia, W., Zhang, C., Chen, J.: An efficient parameterized algorithm for -set packing. J. Algorithms 50, 106–117 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Kovaleva, S., Spieksma, F.C.R.: Approximation algorithms for rectangles tabbing and interval stabbing problems. SIAM J. Discrete Mathematics 20, 748–768 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Lovász, L.: A characterization of perfect graphs. J. Comb. Theory, Ser. B 13, 95–98 (1972)zbMATHCrossRefGoogle Scholar
  28. 28.
    Mahadev, N., Peled, U.: Threshold graphs and related topics, vol. 56. North-Holland, Amsterdam (1995)zbMATHGoogle Scholar
  29. 29.
    Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, USA (2006)zbMATHCrossRefGoogle Scholar
  30. 30.
    Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32, 299–301 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Wagner, K.: Monotonic coverings of finite sets. Elektron. Inform. Kybernet. 20, 633–639 (1984)zbMATHGoogle Scholar
  32. 32.
    Xu, G., Xu, J.: Constant approximation algorithms for rectangle stabbing and related problems. Theory of Computing Systems 40, 187–204 (2007)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Pinar Heggernes
    • 1
  • Dieter Kratsch
    • 2
  • Daniel Lokshtanov
    • 1
  • Venkatesh Raman
    • 3
  • Saket Saurabh
    • 3
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.Université de MetzFrance
  3. 3.The Institute of Mathematical SciencesChennaiIndia

Personalised recommendations