Intuitive Algorithms and t-Vertex Cover

  • Joachim Kneis
  • Daniel Mölle
  • Stefan Richter
  • Peter Rossmanith
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4288)


Many interesting results that improve on the exponential running times of exact algorithms for NP-hard problems have been obtained in recent years. One example that has attracted quite some attention of late is t -Vertex Cover, the problem of finding k nodes that cover at least t edges in a graph. Following the first proof of fixed-parameter tractability, several algorithms for this problem have been presented in rapid succession. We improve on the best known runtime bound, designing and analyzing an intuitive randomized algorithm that takes no more than O(2.0911 t n 4) steps. In fact, we observe and encourage a renewed vigor towards the design of intuitive algorithms within the community. That is, we make a plea to prefer simple, comprehendable, easy-to-implement and easy-to-verify algorithms at the expense of a more involved analysis over more complicated algorithms that are specifically tailored to ease the analysis.


Failure Probability Success Probability Exact Algorithm Vertex Cover Steiner Tree Problem 


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  1. 1.
    Alon, N., Goldreich, O., Håstad, J., Peralta, R.: Simple constructions of almost k-wise independent random variables. Journal of Random structures and Algorithms 3(3), 289–304 (1992)MATHCrossRefGoogle Scholar
  2. 2.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. Journal of the ACM 42(4), 844–856 (1995)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    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. 4.
    Beigel, R.: Finding maximum independent sets in sparse and general graphs. In: Proc. of 10th SODA, pp. 856–857 (1999)Google Scholar
  5. 5.
    Bläser, M.: Computing small partial coverings. Information Processing Letters 85, 327–331 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    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
  7. 7.
    Cai, L., Chan, S.M., Chan, S.O.: Random separation: A new method for solving fixed-cardinality optimization problems. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, Springer, Heidelberg (2006)Google Scholar
  8. 8.
    Cai, L., Chan, S.M., Chan, S.O.: Random separation: A new method for solving fixed-parameter problems (manuscript, 2006)Google Scholar
  9. 9.
    Sunil Chandran, L., Grandoni, F.: Refined memorization for vertex cover. Information Processing Letters 93, 125–131 (2005)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chen, J., Kanj, I.A., Jia, W.: Vertex cover: Further observations and further improvements. Journal of Algorithms 41, 280–301 (2001)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Chen, J., Kanj, I.A., Xia, G.: Simplicity is beauty: Improved upper bounds for vertex cover. Technical Report TR05-008, School of CTI, DePaul University (2005)Google Scholar
  12. 12.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  13. 13.
    Dreyfus, S.E., Wagner, R.A.: The Steiner problem in graphs. Networks 1, 195–207 (1972)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and conquer: Domination – A case study. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, Springer, Heidelberg (2005)Google Scholar
  15. 15.
    Guo, J., Niedermeier, R., Wernicke, S.: Parameterized complexity of generalized vertex cover problems. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 36–48. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Hochbaum, D.S.: The t-vertex cover problem: Extending the half integrality framework with budget constraints. In: Jansen, K., Rolim, J.D.P. (eds.) APPROX 1998. LNCS, vol. 1444, pp. 111–122. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  17. 17.
    Iwama, K., Tamaki, S.: Improved upper bounds for 3-SAT. In: Proc. of 15th SODA, pp. 328–328 (2004)Google Scholar
  18. 18.
    Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Algorithms based on the treewidth of sparse graphs. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 385–396. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Mölle, D., Richter, S., Rossmanith, P.: Enumerate and expand: Improved algorithms for connected vertex cover and tree cover. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 270–280. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. 20.
    Mölle, D., Richter, S., Rossmanith, P.: A faster algorithm for the Steiner tree problem. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 561–570. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    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. 22.
    Niedermeier, R., Rossmanith, P.: On efficient fixed parameter algorithms for Weighted Vertex Cover. Journal of Algorithms 47, 63–77 (2003)MATHMathSciNetGoogle Scholar
  23. 23.
    Petrank, E.: The hardness of approximation: Gap location. Computational Complexity 4, 133–157 (1994)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Raman, V., Saurabh, S.: Triangles, 4-cycles and parameterized (in-) tractability. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, Springer, Heidelberg (2006)CrossRefGoogle Scholar
  25. 25.
    Reingold, O.: Undirected ST-connectivity in Log-Space. In: Proc. of 37th STOC, pp. 376–385 (2005)Google Scholar
  26. 26.
    Robson, J.M.: Algorithms for maximum independent sets. Journal of Algorithms 7, 425–440 (1986)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Schöning, U.: A probabilistic algorithm for k-SAT and constraint satisfaction problems. In: Proc. of 40th FOCS, pp. 410–414 (1999)Google Scholar
  28. 28.
    Scott, A., Sorkin, G.B.: Faster algorithms for Max-CUT and Max-CSP, with polynomial expected time for sparse instances. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 382–395. Springer, Heidelberg (2003)Google Scholar
  29. 29.
    Spielman, D.A., Teng, S.-H.: Smoothed analysis: Why the simplex algorithm usually takes polynomial time. In: Proc. of 33rd STOC, pp. 296–305 (2001)Google Scholar
  30. 30.
    Williams, R.: A new algorithm for optimal constraint satisfaction and its implications. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1227–1237. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Joachim Kneis
    • 1
  • Daniel Mölle
    • 1
  • Stefan Richter
    • 1
  • Peter Rossmanith
    • 1
  1. 1.Department of Computer ScienceRWTH Aachen UniversityFed. Rep. of Germany

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