Algorithmica

, Volume 52, Issue 2, pp 167–176 | Cite as

Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems

  • M. R. Fellows
  • C. Knauer
  • N. Nishimura
  • P. Ragde
  • F. Rosamond
  • U. Stege
  • D. M. Thilikos
  • S. Whitesides
Article

Abstract

We obtain faster algorithms for problems such as r-dimensional matching and r-set packing when the size k of the solution is considered a parameter. We first establish a general framework for finding and exploiting small problem kernels (of size polynomial in k). This technique lets us combine Alon, Yuster and Zwick’s color-coding technique with dynamic programming to obtain faster fixed-parameter algorithms for these problems. Our algorithms run in time O(n+2O(k)), an improvement over previous algorithms for some of these problems running in time O(n+kO(k)). The flexibility of our approach allows tuning of algorithms to obtain smaller constants in the exponent.

Keywords

Parameterized complexity Fixed parameter tractable Graph matching Set packing Color coding 

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References

  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. Assoc. Comput. Mach. 42(4), 844–856 (1995) MATHMathSciNetGoogle Scholar
  2. 2.
    Ausiello, G., D’Atri, A., Protasi, M.: Structure preserving reductions among context optimization problems. J. Comput. Syst. Sci. 21, 136–153 (1980) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chen, J., Friesen, D.K., Jia, W., Kanj, I.: Using nondeterminism to design deterministic algorithms. In: Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science (FST TCS). Lecture Notes in Computer Science, vol. 2245, pp. 120–131. Springer, New York (2001) CrossRefGoogle Scholar
  4. 4.
    Chen, J., Lu, S., Sze, S.-H., Zhang, F.: Improved algorithms for path, matching, and packing problems. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 298–307 (2007) Google Scholar
  5. 5.
    Chor, B., Fellows, M.R., Juedes, D.: Linear kernels in linear time, or how to save k colors in O(n 2) steps. In: Proceedings of the 30th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2004), pp. 257–269 (2004) Google Scholar
  6. 6.
    Fellows, M.R.: Blow-ups, win/win’s, and crown rules: some new directions in FPT. In: Proceedings of the 29th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 03). Lecture Notes in Computer Science, vol. 2880, pp. 1–12. Springer, New York (2003) Google Scholar
  7. 7.
    Fellows, M.R., Heggernes, P., Rosamond, F.A., Sloper, C., Telle, J.A.: Exact algorithms for finding k disjoint triangles in an arbitrary graph. In: Proceedings of the 30th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2004), pp. 257–269 (2004) Google Scholar
  8. 8.
    Fellows, M.R., Knauer, C., Nishimura, N., Ragde, P., Rosamond, F., Stege, U., Thilikos, D.M., Whitesides, S.: Faster fixed-parameter tractable algorithms for matching and packing problems. In: Proceedings of the 12th Annual European Symposium on Algorithms (ESA), pp. 311–322 (2004) Google Scholar
  9. 9.
    Fredman, M.L., Komlós, J., Szemerédi, E.: Storing a sparse table with o(1) worst-case access time. In: Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science, pp. 165–169 (1982) Google Scholar
  10. 10.
    Jia, W., Zhang, C., Chen, J.: An efficient parameterized algorithm for m-set packing. J. Algorithms 50(1), 106–117 (2004) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972) Google Scholar
  12. 12.
    Kneis, J., Moelle, D., Richter, S., Rossmanith, P.: Divide-and-color. In: Proceedings of the 32nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pp. 58–67 (2006) Google Scholar
  13. 13.
    Koutis, I.: A faster parameterized algorithm for set packing. Inf. Process. Lett. 94, 7–9 (2005) CrossRefMathSciNetGoogle Scholar
  14. 14.
    Liu, Y., Lu, S., Chen, J., Sze, S.: Greedy localization and color-coding: improved matching and packing algorithms. In: Proceedings of the 2nd International Workshop on Parameterized and Exact Computation. Lecture Notes in Computer Science, vol. 4162, pp. 84–95. Springer, New York (2006) CrossRefGoogle Scholar
  15. 15.
    Marx, D.: Parameterized complexity of constraint satisfaction problems. In: Proceedings of the 19th Annual IEEE Conference on Computational Complexity, 2004 Google Scholar
  16. 16.
    Slot, C.F., van Emde Boas, P.: On tape versus core; an application of space efficient perfect hash functions to the invariance of space. Elektron. Inf. Kybern. 21(4/5), 246–253 (1985) Google Scholar
  17. 17.
    Woeginger, G.J.: Exact algorithms for NP-hard problems, a survey. In: Combinatorial Optimization—Eureka, You Shrink!. Lecture Notes on Computer Science, vol. 2570, pp. 185–207. Springer, New York (2003) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • M. R. Fellows
    • 1
  • C. Knauer
    • 2
  • N. Nishimura
    • 3
  • P. Ragde
    • 3
  • F. Rosamond
    • 1
  • U. Stege
    • 4
  • D. M. Thilikos
    • 5
  • S. Whitesides
    • 6
  1. 1.School of Electrical Engineering and Computer ScienceUniversity of NewcastleNewcastleAustralia
  2. 2.Institute of Computer ScienceFreie Universität BerlinBerlinGermany
  3. 3.School of Computer ScienceUniversity of WaterlooWaterlooCanada
  4. 4.Department of Computer ScienceUniversity of VictoriaVictoriaCanada
  5. 5.Departament de Llenguatges i Sistemes InformàticsUniversitat Politècnica de CatalunyaBarcelonaSpain
  6. 6.School of Computer ScienceMcGill UniversityMontrealCanada

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