Parameterized Complexity of Generalized Vertex Cover Problems

  • Jiong Guo
  • Rolf Niedermeier
  • Sebastian Wernicke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3608)

Abstract

Important generalizations of the Vertex Cover problem (Connected Vertex Cover, Capacitated Vertex Cover, and Maximum Partial Vertex Cover) have been intensively studied in terms of approximability. However, their parameterized complexity has so far been completely open. We close this gap here by showing that, with the size of the desired vertex cover as parameter, Connected Vertex Cover and Capacitated Vertex Cover are both fixed-parameter tractable while Maximum Partial Vertex Cover is W[1]-hard. This answers two open questions from the literature. The results extend to several closely related problems. Interestingly, although the considered generalized Vertex Cover problems behave very similar in terms of constant-factor approximability, they display a wide range of different characteristics when investigating their parameterized complexities.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abu-Khzam, F.N., Collins, R.L., Fellows, M.R., Langston, M.A., Suters, W.H., Symons, C.T.: Kernelization algorithms for the Vertex Cover problem: theory and experiments. In: Proc. ALENEX 2004, pp. 62–69. ACM/SIAM (2004)Google Scholar
  2. 2.
    Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for Dominating Set and related problems on planar graphs. Algorithmica 33(4), 461–493 (2002)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alber, J., Dorn, F., Niedermeier, R.: Experimental evaluation of a tree decomposition based algorithm for Vertex Cover on planar graphs. Discrete Applied Mathematics 145(2), 219–231 (2005)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Alber, J., Fernau, H., Niedermeier, R.: Parameterized complexity: exponential speed-up for planar graph problems. Journal of Algorithms 52, 26–56 (2004)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Alber, J., Gramm, J., Niedermeier, R.: Faster exact algorithms for hard problems: a parameterized point of view. Discrete Mathematics 229, 3–27 (2001)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Arkin, E.M., Halldórsson, M.M., Hassin, R.: Approximating the tree and tour covers of a graph. Information Processing Letters 47(6), 275–282 (1993)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Arvind, V., Raman, V.: Approximation algorithms for some parameterized counting problems. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 453–464. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Bläser, M.: Computing small partial coverings. Information Processing Letters 85(6), 327–331 (2003)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bshouty, N.H., Burroughs, L.: Massaging a linear programming solution to give a 2-approximation for a generalization of the Vertex Cover problem. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 298–308. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  11. 11.
    Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. Journal of Computer and System Sciences 67(4), 789–807 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chandran, L.S., Grandoni, F.: Refined memorisation for Vertex Cover. Information Processing Letters 93(3), 125–131 (2005)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Cheetham, J., Dehne, F., Rau-Chaplin, A., Stege, U., Taillon, P.J.: Solving large FPT problems on coarse-grained parallel machines. Journal of Computer and System Sciences 67(4), 691–706 (2003)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Chen, J., Kanj, I.A., Jia, W.: Vertex Cover: further observations and further improvements. Journal of Algorithms 41, 280–301 (2001)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Chuzhoy, J., Naor, J.S.: Covering problems with hard capacities. In: Proc. 43rd FOCS, pp. 481–489. IEEE Computer Society Press, Los Alamitos (2002)Google Scholar
  16. 16.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)MATHGoogle Scholar
  17. 17.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  18. 18.
    Dreyfus, S.E., Wagner, R.A.: The Steiner problem in graphs. Networks 1, 195–207 (1972)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Fellows, M.R.: New directions and new challenges in algorithm design and complexity, parameterized. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 505–520. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  20. 20.
    Gandhi, R., Halperin, E., Khuller, S., Kortsarz, G., Srinivasan, A.: An improved approximation algorithm for Vertex Cover with hard capacities. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 164–175. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  21. 21.
    Gandhi, R., Khuller, S., Srinivasan, A.: Approximation algorithms for partial covering problems. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 225–236. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  22. 22.
    Guha, S., Hassin, R., Khuller, S., Or, E.: Capacitated vertex covering. Journal of Algorithms 48(1), 257–270 (2003)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Könemann, J., Konjevod, G., Parekh, O., Sinha, A.: Improved approximations for tour and tree covers. Algorithmica 38(3), 441–449 (2004)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Niedermeier, R.: Ubiquitous parameterization—invitation to fixed-parameter algorithms. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 84–103. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  25. 25.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, forthcoming (2005)Google Scholar
  26. 26.
    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
  27. 27.
    Niedermeier, R., Rossmanith, P.: A general method to speed up fixed-parametertractable algorithms. Information Processing Letters 73, 125–129 (2000)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Niedermeier, R., Rossmanith, P.: On efficient fixed-parameter algorithms for Weighted Vertex Cover. Journal of Algorithms 47(2), 63–77 (2003)MATHMathSciNetGoogle Scholar
  29. 29.
    Nishimura, N., Ragde, P., Thilikos, D.M.: Fast fixed-parameter tractable algorithms for nontrivial generalizations of Vertex Cover. In: Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 2001. LNCS, vol. 2125, pp. 75–86. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  30. 30.
    Prieto, E., Sloper, C.: Either/or: using vertex cover structure in designing FPTalgorithms—the case of k-Internal Spanning Tree. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 474–483. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jiong Guo
    • 1
  • Rolf Niedermeier
    • 1
  • Sebastian Wernicke
    • 1
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaFed. Rep. of Germany

Personalised recommendations