Advertisement

Exponentialzeit-Algorithmen für Färbbarkeitsprobleme

  • Frank Gurski
  • Irene Rothe
  • Jörg Rothe
  • Egon Wanke
Chapter
Part of the eXamen.press book series (EXAMEN)

Zusammenfassung

Zunächst stellen wir in diesem Abschnitt ein paar Algorithmen vor, die auf einfachen Ideen beruhen und den naiven Algorithmus für Dreifärbbarkeit bereits schlagen (auch wenn sie natürlich immer noch Exponentialzeit brauchen; schließlich ist das Dreifärbbarkeitsproblem nach Satz 5.26 NP-vollständig). Anschließend gehen wir kurz auf die Motivation für exakte Exponentialzeit-Algorithmen ein und erläutern, weshalb solche Verbesserungen für praktische Anwendungen sehr sinnvoll sein können.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literaturverzeichnis

  1. [Woe03]
    G. Woeginger. Exact algorithms for NP-hard problems. In M.Jünger, G. Reinelt, and G. Rinaldi, editors, Combinatorical Optimization: “Eureka, you shrink!”, pages 185–207. Springer-Verlag Lecture Notes in Computer Science #2570, 2003.Google Scholar
  2. [TT77]
    R. Tarjan and A. Trojanowski. Finding a maximum independent set. SIAM Journal on Computing, 6(3):537–546, 1977.MATHCrossRefMathSciNetGoogle Scholar
  3. [GK04]
    V. Guruswami and S. Khanna. On the hardness of 4-coloring a 3-colorable graph. SIAM Journal on Discrete Mathematics, 18(1):30–40, 2004.MATHCrossRefMathSciNetGoogle Scholar
  4. [Epp04]
    D. Eppstein. Quasiconvex analysis of backtracking algorithms. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 788–797. Society for Industrial and Applied Mathematics, January 2004.Google Scholar
  5. [Hol81]
    I. Holyer. The NP-completeness of edge-coloring. SIAM Journal on Computing, 10(4):718–720, 1981.MATHCrossRefMathSciNetGoogle Scholar
  6. [RR06]
    T. Riege and J. Rothe. Improving deterministic and randomized exponential-time algorithms for the satisfiability, the colorability, and the domatic number problem. Journal of Universal Computer Science, 12(6):725–745, 2006.MathSciNetGoogle Scholar
  7. [Sch05]
    U. Schöning. Algorithmics in exponential time. In Proceedings of the 22nd Annual Symposium on Theoretical Aspects of Computer Science, pages 36–43. Springer-Verlag Lecture Notes in Computer Science #3404, February 2005.Google Scholar
  8. [Sch96]
    I. Schiermeyer. Fast exact colouring algorithms. In Tatra Mountains Mathematical Publications, volume9, pages 15–30, 1996.MATHMathSciNetGoogle Scholar
  9. [KLS00]
    S. Khanna, N. Linial, and S. Safra. On the hardness of approximating the chromatic number. Combinatorica, 20(3):393–415, 2000.MATHCrossRefMathSciNetGoogle Scholar
  10. [BE05]
    R. Beigel and D. Eppstein. 3-coloring in time \({{\mathcal{O}}}(1.3289^n\)). Journal of Algorithms, 54(2):168–204, 2005.MATHCrossRefMathSciNetGoogle Scholar
  11. [PW89]
    A. Petford and D. Welsh. A randomised 3-colouring algorithm. Discrete Mathematics, 74(1–2):253–261, 1989.MATHCrossRefMathSciNetGoogle Scholar
  12. [Epp01]
    D. Eppstein. Improved algorithms for 3-coloring, 3-edge-coloring, and constraint satisfaction. In Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 329–337. Society for Industrial and Applied Mathematics, January 2001.Google Scholar
  13. [Rob86]
    J. Robson. Algorithms for maximum independent sets. Journal of Algorithms, 7(3):425–440, 1986.MATHCrossRefMathSciNetGoogle Scholar
  14. [MM65]
    J. Moon and L. Moser. On cliques in graphs. Israel Journal of Mathematics, 3:23–28, 1965.MATHCrossRefMathSciNetGoogle Scholar
  15. [BK97]
    A. Blum and D. Karger. An \(\tilde{{\cal O}}(n^{\frac{3}{14}})\)-coloring algorithm for 3-colorable graphs. Information Processing Letters, 61(1):49–53, 1997.CrossRefMathSciNetGoogle Scholar
  16. [BE95]
    R. Beigel and D. Eppstein. 3-coloring in time \({{\mathcal{O}}}(1.3446^n\)): A no-MIS algorithm. In Proceedings of the 36th IEEE Symposium on Foundations of Computer Science, pages 444–452. IEEE Computer Society Press, October 1995.Google Scholar
  17. [PU59]
    M. Paull and S. Unger. Minimizing the number of states in incompletely specified state machines. IRE Transactions on Electronic Computers, EC-8:356–367, 1959.CrossRefGoogle Scholar
  18. [CM87]
    J. Cai and G. Meyer. Graph minimal uncolorability is DP-complete. SIAM Journal on Computing, 16(2):259–277, 1987.MATHCrossRefMathSciNetGoogle Scholar
  19. [Wag87]
    K. Wagner. More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science, 51:53–80, 1987.MATHCrossRefMathSciNetGoogle Scholar
  20. [Rot00]
    J. Rothe. Heuristics versus completeness for graph coloring. Chicago Journal of Theoretical Computer Science, vol.2000, article1:1–16, February 2000.Google Scholar
  21. [AK97]
    N. Alon and N. Kahale. A spectral technique for coloring random 3-colorable graphs. SIAM Journal on Computing, 26(6):1733–1748, 1997.MATHCrossRefMathSciNetGoogle Scholar
  22. [Sch93]
    I. Schiermeyer. Deciding 3-colourability in less than \({{\mathcal{O}}}(1.415^n)\) steps. In Proceedings of the 19th International Workshop on Graph-Theoretic Concepts in Computer Science, pages 177–182. Springer-Verlag Lecture Notes in Computer Science #790, June 1993.Google Scholar
  23. [Epp03]
    D. Eppstein. Small maximal independent sets and faster exact graph coloring. Journal of Graph Algorithms and Applications, 7(2):131–140, 2003.MATHMathSciNetGoogle Scholar
  24. [Rot03]
    J. Rothe. Exact complexity of Exact-Four-Colorability. Information Processing Letters, 87(1):7–12, 2003.Google Scholar
  25. [Vla95]
    R. Vlasie. Systematic generation of very hard cases for 3-colorability. In Proceedings of the 7th IEEE International Conference on Tools with Artificial Intelligence, pages 114–119. IEEE, August/September 1995.Google Scholar
  26. [Bys02]
    J. Byskov. Chromatic number in time \(\mathcal{O}(2.4023^n)\) using maximal independent sets. Technical Report RS-02-45, Center for Basic Research in Computer Science (BRICS), December 2002.Google Scholar
  27. [Law76]
    E. Lawler. A note on the complexity of the chromatic number problem. Information Processing Letters, 5(3):66–67, 1976.MATHCrossRefMathSciNetGoogle Scholar
  28. [JPY88]
    D. Johnson, C. Papadimitriou, and M. Yannakakis. On generating all maximal independent sets. Information Processing Letters, 27(3):119–123, 1988.MATHCrossRefMathSciNetGoogle Scholar
  29. [Rob01]
    J. Robson. Finding a maximum independent set in time \({{\mathcal{O}}}(2^{\frac{n}{4}})\). Technical Report TR 1251-01, LaBRI, Université Bordeaux I, 2001. Available on-line at http://dept-info.labri.fr/∼robson/mis/techrep.html.
  30. [Sch99]
    U. Schöning. A probabilistic algorithm for k-SAT and constraint satisfaction problems. In Proceedings of the 40th IEEE Symposium on Foundations of Computer Science, pages 410–414. IEEE Computer Society Press, October 1999.Google Scholar
  31. [Ada79]
    D. Adams. The Hitchhiker’s Guide to the Galaxy. Pan Books, 1979.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Frank Gurski
    • 1
  • Irene Rothe
    • 2
  • Jörg Rothe
    • 1
  • Egon Wanke
    • 1
  1. 1.Institut für InformatikHeinrich-Heine-Universität DüsseldorfDüsseldorfDeutschland
  2. 2.Fachbereich für Maschinenbau Elektrotechnik und TechnikjournalismusHochschule Bonn-Rhein-SiegSankt AugustinDeutschland

Personalised recommendations