Either/Or: Using Vertex Cover Structure in Designing FPT-Algorithms — the Case of k-Internal Spanning Tree

  • Elena Prieto
  • Christian Sloper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2748)


To determine if a graph has a spanning tree with few leaves is NP-hard. In this paper we study the parametric dual of this problem, k-Internal Spanning Tree (Does G have a spanning tree with at least k internal vertices?). We give an algorithm running in time O(24klogk ·k 7/2 + k 2 ·n 2). We also give a 2-approximation algorithm for the problem.

However, the main contribution of this paper is that we show the following remarkable structural bindings between k-Internal Spanning Tree and k-Vertex Cover:
  • No for k-Vertex Cover implies Yes for k-Internal Spanning Tree.

  • Yes for k-Vertex Cover implies No for (2k+1)-Internal Spanning Tree.

We give a polynomial-time algorithm that produces either a vertex cover of size kor a spanning tree with at least k internal vertices. We show how to use this inherent vertex cover structure to design algorithms for FPT problems, here illustrated mainly by k-Internal Spanning Tree. We also briefly discuss the application of this vertex cover methodology to the parametric dual of the Dominating Set problem. This design technique seems to apply to many other FPT problems.


Spanning trees fixed-parameter tractability vertex cover kernelization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A03]
    Abu-Khzam, F.: Private communication Google Scholar
  2. [AN02]
    Alber, J., Niedermeier, R.: Improved tree decomposition based algorithms for domination-like problems. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 613–627. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. [BFR98]
    Balasubramanian, R., Fellows, M.R., Raman, V.: An Improved Fixed Parameter Algorithm for Vertex Cover. Information Processing Letters 65(3), 163–168 (1998)CrossRefMathSciNetGoogle Scholar
  4. [CFJ03]
    Chor, B., Fellows, M., Juedes, D.: Private communication concerning (manuscript) (in preparation) Google Scholar
  5. [CCDF97]
    Cai, L., Chen, J., Downey, R., Fellows, M.: The parameterized complexity of short computation and factorization. Archive for Mathematical Logic 36, 321–338 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [CKJ01]
    Chen, J., Kanj, I., Jia, W.: Vertex cover: Further Observations and Further Improvements. Journal of Algorithms 41, 280–301 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [CLR90]
    Cormen, T.H., Leierson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press, CambridgeGoogle Scholar
  8. [DF95a]
    Downey, R., Fellows, M.: Parameterized Computational Feasibility. In: Clote, P., Remmel, J. (eds.) Feasible Mathematics II, pp. 219–244. Birkhauser, Boston (1995)Google Scholar
  9. [DF95b]
    Downey, R., Fellows, M.: Fixed-parameter tractability and completeness II: completeness for W[1]. Theoretical Computer Science A 141, 109–131 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  10. [DF98]
    Downey, R., Fellows, M.: Parameterized Complexity. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  11. [DFS99]
    Downey, R., Fellows, M., Stege, U.: Parameterized complexity: a framework for systematically confronting computational intractability. In: Graham, R., Kratochvil, J., Nesetril, J., Roberts, F. (eds.) Contemporary Trends in Discrete Mathematics. AMS-DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 49, pp. 49–99 (1999)Google Scholar
  12. [FFLR03]
    Faisal, A., Fellows, M., Langston, M., Rosamond, F.: Private communication concerning (manuscript) (in preparation) Google Scholar
  13. [FMRS01]
    Fellows, M., McCartin, C., Rosamond, F., Stege, U.: Spanning Trees with Few and Many Leaves (to appear)Google Scholar
  14. [GMM94]
    Galbiati, G., Maffioli, F., Morzenti, A.: A Short Note on the Approximability of the Maximum Leaves Spanning Tree Problem. Information Processing Letters 52, 45–49 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  15. [GMM97]
    Galbiati, G., Morzenti, A., Maffioli, F.: On the Approximability of some Maximum Spanning Tree Problems. Theoretical Computer Science 181, 107–118 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  16. [GJ79]
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  17. [KR00]
    Khot, S., Raman, V.: Parameterized Complexity of Finding Hereditary Properties. In: Du, D.-Z., Eades, P., Sharma, A.K., Lin, X., Estivill-Castro, V. (eds.) COCOON 2000. LNCS, vol. 1858, p. 137. Springer, Heidelberg (2000); Theoretical Computer Science (COCOON 2000 special issue)CrossRefGoogle Scholar
  18. [L03]
    Langston, M.: Private communication Google Scholar
  19. [LR98]
    Lu, H.-I., Ravi, R.: Approximating Maximum Leaf Spanning Trees in Almost Linear Time. Journal of Algorithms 29, 132–141 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  20. [McC03]
    McCartin, C.: Ph.D. dissertation in Computer Science, Victoria University, Wellington, New Zealand (2003) Google Scholar
  21. [NR99b]
    Niedermeier, R., Rossmanith, P.: Upper Bounds for Vertex Cover Further Improved. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 561–570. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  22. [PT93]
    Telle, J.A., Proskurowski, A.: Practical algorithms on partial k-trees with an application to domination-like problems. In: Dehne, F., Sack, J.-R., Santoro, N. (eds.) WADS 1993. LNCS, vol. 709, pp. 610–621. Springer, Heidelberg (1993)Google Scholar
  23. [RS99]
    Robertson, N., Seymor, P.D.: Graph Minors. XX Wagner’s conjecture (to appear) Google Scholar
  24. [Ste00]
    Stege, U.: Ph.D. dissertation in Computer Science, ETH, Zurich, Switzerland (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Elena Prieto
    • 1
  • Christian Sloper
    • 2
  1. 1.School of Electrical Engineering and Computer ScienceThe University of NewcastleAustralia
  2. 2.Department of InformaticsUniversity of BergenNorway

Personalised recommendations