Fast Fixed-Parameter Tractable Algorithms for Nontrivial Generalizations of Vertex Cover

  • Naomi Nishimura
  • Prabhakar Ragde
  • Dimitrios M. Thilikos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2125)


Our goal in this paper is the development of fast algorithms for recognizing general classes of graphs. We seek algorithms whose complexity can be expressed as a linear function of the graph size plus an exponential function of k, a natural parameter describing the class. Our classes are of the form \( \mathcal{W}_k (\mathcal{G}) \), graphs that can be formed by augmenting graphs in \( \mathcal{G} \) by adding at most k vertices (and incident edges). If \( \mathcal{G} \) is the class of edgeless graphs, \( \mathcal{W}_k (\mathcal{G}) \) is the class of graphs with a vertex cover of size at most k.

We describe a recognition algorithm for \( \mathcal{W}_k (\mathcal{G}) \) running in time O((g + k)|V(G)| + (fk) k), where g and f are modest constants depending on the class \( \mathcal{G} \), when \( \mathcal{G} \) is a minor-closed class such that each graph in G has bounded maximum degree, and all obstructions of G (minor-minimal graphs outside \( \mathcal{G} \)) are connected. If \( \mathcal{G} \) is the class of graphs with maximum degree bounded by D (not closed under minors), we can still recognize graphs in \( \mathcal{W}_k (\mathcal{G}) \) in time O(|V(G)|(D + k) + k(D + k) k+3).

Our results are obtained by considering minor-closed classes \( \mathcal{G} \) for which all obstructions are connected graphs, and showing that the size of any obstruction for \( \mathcal{W}_k (\mathcal{G}) \) is O(tk 7 + t 7 k 2), where t is a bound on the size of obstructions for \( \mathcal{G} \).


Vertex Cover Graph Class Graph Minor Bounded Degree Proper Interval Graph 
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. Arn85.
    Stefan Arnborg. Efficient algorithms for combinatorial problems on graphs with bounded decomposability-A survey. BIT, 25:2–23, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  2. BFR98.
    R. Balasubramanian, Michael Fellows, and Venkatesh Raman. An improved fixed-parameter algorithm for vertex cover. Inform. Proc. Letters, 65:163–168, 1998.CrossRefMathSciNetGoogle Scholar
  3. BG93.
    Jonathan F. Buss and Judy Goldsmith. Nondeterminism within P. SIAM J. Computing, 22:560–572, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  4. BRST91.
    Daniel Bienstock, Neil Robertson, Paul D. Seymour, and Robin Thomas. Quickly excluding a forest. J. Comb. Theory Series B, 52:274–283, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  5. Cai96.
    Leizhen Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Inform. Proc. Letters, 58:171–176, 1996.zbMATHCrossRefGoogle Scholar
  6. CCDF97.
    Liming Cai, Jianer Chen, Rodney Downey, and Michael Fellows. Advice classes of parameterized tractability. Annals of Pure and Applied Logic, 84:119–138, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  7. CKJ99.
    J. Chen, I.A. Kanj, and W. Jia. Vertex cover: Further observations and further improvements. In Proceedings of the 25th International Workshop on Graph-Theoretical Concepts in Computer Science, pages 313–324. Springer-Verlag, 1999.Google Scholar
  8. DF95.
    Rodney G. Downey and Michael R. Fellows. Fixed-parameter tractability and completeness I: Basic results. SIAM J. Comput., 24:873–921, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  9. DF99.
    Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Springer-Verlag, 1999.Google Scholar
  10. DFS99.
    Rodney G. Downey, Michael R. Fellows, and Ulrike Stege. Computational tractability: the view from Mars. Bulletin of the European Association of Theoretical Computer Science, 69:73–97, 1999.zbMATHMathSciNetGoogle Scholar
  11. Din95.
    Michael J. Dinneen. Bounded Combinatorial Width and Forbidden Substructures. PhD thesis, Department of Computer Science, University of Victoria, 1995.Google Scholar
  12. Din97.
    Michael J. Dinneen. Too many minor order obstructions (for parametrized lower ideals). Journal of Universal Computer Science, 3(11):1199–1206, 1997.zbMATHMathSciNetGoogle Scholar
  13. FL88.
    Michael R. Fellows and Michael A. Langston. Nonconstructive tools for proving polynomial-time decidability. J. ACM, 35:727–739, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  14. FL94.
    Michael R. Fellows and Michael A. Langston. On search, decision, and the efficiency of polynomial-time algorithms. J. Comp. Syst. Sc., 49(3):769–779, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  15. FPT95.
    Martin Farach, Teresa Przytycka, and Mikkel Thorup. On the agreement of many trees. Inform. Proc. Letters, 55:297–301, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  16. FRS87.
    Harvey Friedman, Neil Robertson, and Paul D. Seymour. The metamathe-matics of the graph minor theorem. Contemporary Mathematics, 65:229–261, 1987.MathSciNetGoogle Scholar
  17. KST99.
    Haim Kaplan, Ron Shamir, and Robert E. Tarjan. Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM J. Comput., 28(5):1906–1922, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  18. Lag93.
    Jens Lagergren. An upper bound on the size of an obstruction. In Neil Robertson and Paul Seymour, editors, Graph Structure Theory, Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference, Seattle WA, June 1991, pages 601–621, Providence, RI, 1993. American Math. Soc. Contemp. Math. 147.Google Scholar
  19. Lag98.
    Jens Lagergren. Upper bounds on the size of obstructions and intertwines. J. Combin. Theory Ser. B, 73:7–40, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  20. LP80.
    J. M. Lewis and Christos H. Papadimitriou. The node-deletion problem for hereditary properties is NP-complete. J. Comp. Syst. Sc., 20:219–230, 1980.zbMATHCrossRefGoogle Scholar
  21. LP98.
    Michael A Langston and Barbara C. Plaut. On algorithmic applications of the immersion order. An overview of ongoing work presented at the Third Slovenian International Conference on Graph Theory. Discrete Mathematics, 182(1–3):191–196, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  22. Meh84.
    Kurt Mehlhorn. Graph Algorithms and NP-Completeness, volume 2 of Data Structures and Algorithms. Springer Verlag, Berlin, 1984.zbMATHGoogle Scholar
  23. MR99.
    Meena Mahajan and Venkatesh Raman. Parameterizing above guaranteed values: maxsat and maxcut. J. Algorithms, 31(2):335–354, May 1999.Google Scholar
  24. Nie98.
    Rolf Niedermeier. Some prospects for efficient fixed parameter algorithms. In Proceedings of the 25th Conference on Current Trends in Theory and Practice of Informatics (SOFSEM’98), volume 1521 of Lecture Notes in Computer Science, pages 168–185, 1998.Google Scholar
  25. NR99.
    Rolf Niedermeier and Peter Rossmanith. Upper bounds for vertex cover further improved. In C. Meinel, S. Tison, editors, Proceedings of the 16th Symposium on Theoretical Aspects of Computer Science (STACS’99), volume 1563 of Lecture Notes in Computer Science, pages 561–570, 1999.Google Scholar
  26. Ram95.
    Siddharthan Ramachandramurthi. A lower bound for treewidth and its consequences. In Ernst W. Mayr, Gunther Schmidt, and Gottfried Tinhofer, editors, Proceedings 20th International Workshop on Graph Theoretic Concepts in Computer Science WG’94, volume 903 of Lecture Notes in Computer Science, pages 14–25. Springer Verlag, 1995.Google Scholar
  27. RS85a.
    Neil Robertson and Paul D. Seymour. Disjoint paths-A survey. SIAM J. Alg. Disc. Meth., 6:300–305, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  28. RS85b.
    Neil Robertson and Paul D. Seymour. Graph minors — A survey. In I. Anderson, editor, Surveys in Combinatorics, pages 153–171. Cambridge Univ. Press, 1985.Google Scholar
  29. RS95.
    Neil Robertson and Paul D. Seymour. Graph minors. XIII. The disjoint paths problem. J. Comb. Theory Series B, 63:65–110, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  30. Thi00.
    Dimitrios M. Thilikos. Algorithms and obstructions for linear-width and related search parameters. Discrete Applied Mathematics, 105:239–271, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  31. TUK94.
    Atsushi Takahashi, Shuichi Ueno, and Yoji Kajitani. Minimal acyclic forbidden minors for the family of graphs with bounded path-width. Discrete Mathematics, 127(1/3):293–304, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  32. van90.
    Jan van Leeuwen. Graph algorithms. In Handbook of Theoretical Computer Science, A: Algorithms and Complexity Theory, pages 527–631, Amsterdam, 1990. North Holland Publ. Comp.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Naomi Nishimura
    • 1
  • Prabhakar Ragde
    • 1
  • Dimitrios M. Thilikos
    • 2
  1. 1.Department of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Departament de Llenguatges i Sistemes InformáticsUniversitat Politécnica de CatalunyaBarcelonaSpain

Personalised recommendations