Greedy Localization, Iterative Compression, and Modeled Crown Reductions: New FPT Techniques, an Improved Algorithm for Set Splitting, and a Novel 2k Kernelization for Vertex Cover

  • Frank Dehne
  • Mike Fellows
  • Frances Rosamond
  • Peter Shaw
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3162)


The two objectives of this paper are: (1) to articulate three new general techniques for designing FPT algorithms, and (2) to apply these to obtain new FPT algorithms for Set Splitting and Vertex Cover. In the case of Set Splitting, we improve the best previous \({\mathcal O}^*(72^k)\) FPT algorithm due to Dehne, Fellows and Rosamond [DFR03], to \({\mathcal O}^*(8^k)\) by an approach based on greedy localization in conjunction with modeled crown reduction. In the case of Vertex Cover, we describe a new approach to 2k kernelization based on iterative compression and crown reduction, providing a potentially useful alternative to the Nemhauser-Trotter 2k kernelization.


Vertex Cover Auxiliary Graph Vertex Cover Problem Compression Form Graph Theoretic Concept 
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. [ACFLSS04]
    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: Proceedings ALENEX 2004, ACM/SIAM (2004)Google Scholar
  2. [AS00]
    Ageev, A.A., Sviridenko, M.I.: An approximation algorithm for hypergraph max k-cut with given sizes of parts. In: Paterson, M. (ed.) ESA 2000. LNCS, vol. 1879, pp. 32–41. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. [AYZ95]
    Alon, N., Yuster, R., Zwick, U.: Color-Coding. Journal of the ACM 42, 844–856 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [AE97]
    Andersson, G., Engebretsen, L.: Better approximation algorithms for set splitting and Not-All-Equal SAT. Information Processing Letters 65, 305–311 (1998)CrossRefMathSciNetGoogle Scholar
  5. [CFJ04]
    Chor, B., Fellows, M., Juedes, D.: Linear Kernels in Linear Time, or How to Save k Colors in O(n2) Steps). In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 257–269. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. [CFJK01]
    Chen, J., Friesen, D.K., Jia, W., Kanj, I.A.: Using Nondeterminism to Design Efficient Deterministic Algorithms. In: Hariharan, R., Mukund, M., Vinay, V. (eds.) FSTTCS 2001. LNCS, vol. 2245, pp. 120–131. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. [DFR03]
    Dehne, F., Fellows, M., Rosamond, F.: An FPT Algorithm for Set Splitting. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 180–191. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. [DF99]
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  9. [F03]
    Fellows, M.: Blow-ups, Win/Win’s and Crown Rules: Some New Directions in FPT. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 1–12. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. [FKN04]
    Fellows, M., Knauer, C., Nishimura, N., Ragde, P., Rosamond, F., Stege, U., Thilikos, D., Whitesides, S.: Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 311–322. Springer, Heidelberg (2004) (to appear)CrossRefGoogle Scholar
  11. [FHRST04]
    Fellows, M., Heggernes, P., Rosamond, F., Sloper, C., Telle, J.A.: Exact Algorithms for Finding k Disjoint Triangles in an Arbitrary Graph. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 235–244. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. [GJ79]
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)zbMATHGoogle Scholar
  13. [JZC03]
    Jia, W., Zhang, C., Chen, J.: An Efficient Parameterized Algorithm for Set Packing. To appear in Journal of Algorithms (2003) (manuscript)Google Scholar
  14. [Ma04]
    Marx, D.: Chordal Deletion is Fixed-Parameter Tractable (2004) (manuscript)Google Scholar
  15. [MPS04]
    Mathieson, L., Prieto, E., Shaw, P.: Packing Edge Disjoint Triangles: A Parameterized View. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 127–137. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. [Nie02]
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms, Habilitationschrift, University of Tubingen (2002)Google Scholar
  17. [Pe94]
    Petrank, E.: The hardness of approximation: Gap location. Computational Complexity 4, 133–157 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  18. [PS03]
    Prieto, E., Sloper, C.: Either/Or: Using Vertex Cover Structure in Designing FPT Algorithms–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
  19. [PS04]
    Prieto, E., Sloper, C.: Looking at the Stars. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 138–148. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  20. [RSV03]
    Reed, B., Smith, K., Vetta, A.: Finding Odd Cycle Transversals. Operations Research Letters 32, 299–301 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  21. [Woe03]
    Woeginger, G.J.: Exact Algorithms for NP-Hard Problems: A Survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 184–207. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  22. [ZL01]
    Zhang, H., Ling, C.X.: An Improved Learning Algorithm for Augmented Naive Bayes. In: Cheung, D., Williams, G.J., Li, Q. (eds.) PAKDD 2001. LNCS (LNAI), vol. 2035, pp. 581–586. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Frank Dehne
    • 1
  • Mike Fellows
    • 2
  • Frances Rosamond
    • 2
  • Peter Shaw
    • 2
  1. 1.Griffith UniversityBrisbaneAustralia
  2. 2.University of NewcastleCallaghanAustralia

Personalised recommendations