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The Parameterized Complexity of Some Minimum Label Problems

  • Michael R. Fellows
  • Jiong Guo
  • Iyad A. Kanj
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5911)

Abstract

We study the parameterized complexity of several minimum label graph problems, in which we are given an undirected graph whose edges are labeled, and a property Π, and we are asked to find a subset of edges satisfying property Π that uses the minimum number of labels. These problems have a lot of applications in networking. We show that all the problems under consideration are W[2]-hard when parameterized by the number of used labels, and that they remain W[2]-hard even on graphs whose pathwidth is bounded above by a small constant. On the positive side, we prove that most of these problems are FPT when parameterized by the solution size, that is, the size of the sought edge set. For example, we show that computing a maximum matching or an edge dominating set that uses the minimum number of labels, is FPT when parameterized by the solution size. Proving that some of these problems are FPT is nontrivial, and requires interesting and elegant algorithmic methods that we develop in this paper.

Keywords

Bipartite Graph Partial Solution Parameterized Complexity Edge Incident Span Tree Problem 
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.

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References

  1. 1.
    Broersma, H., Li, X.: Spanning trees with many or few colors in edge-colored graphs. Discussiones Mathematicae Graph Theory 17(2), 259–269 (1997)MathSciNetMATHGoogle Scholar
  2. 2.
    Broersma, H., Li, X., Woeginger, G., Zhang, S.: Paths and cycles in colored graphs. Australasian Journal on Combinatorics 31, 299–311 (2005)MathSciNetMATHGoogle Scholar
  3. 3.
    Brüggemann, T., Monnot, J., Woeginger, G.: Local search for the minimum label spanning tree problem with bounded color classes. Operations Research Letters 31(3), 195–201 (2003)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Carr, R., Doddi, S., Konjevod, G., Marathe, M.: On the red-blue set cover problem. In: Proc. 11th ACM-SIAM SODA, pp. 345–353 (2000)Google Scholar
  5. 5.
    Chang, R., Leu, S.: The minimum labeling spanning trees. IPL 63(5), 277–282 (1997)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 2nd edn. McGraw-Hill Book Company, Boston (2001)MATHGoogle Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Monographs in Computer Science. Springer, New York (1999)Google Scholar
  8. 8.
    Hassin, R., Monnot, J., Segev, D.: Approximation algorithms and hardness results for labeled connectivity problems. Journal of Combinatorial Optimization 14(4), 437–453 (2007)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Jha, S., Sheyner, O., Wing, J.: Two formal analyses of attack graphs. In: Proc. 15th IEEE Computer Security Foundations Workshop, pp. 49–63 (2002)Google Scholar
  10. 10.
    Krumke, S., Wirth, H.: On the minimum label spanning tree problem. IPL 66(2), 81–85 (1998)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Monnot, J.: The labeled perfect matching in bipartite graphs. IPL 96(3), 81–88 (2005)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Uno, T.: Algorithms for enumerating all perfect, maximum and maximal matchings in bipartite graphs. In: Leong, H.-V., Jain, S., Imai, H. (eds.) ISAAC 1997. LNCS, vol. 1350, pp. 92–101. Springer, Heidelberg (1997)Google Scholar
  13. 13.
    Voss, S., Cerulli, R., Fink, A., Gentili, M.: Applications of the pilot method to hard modifications of the minimum spanning tree problem. In: Proc. 18th MINI EURO Conference on VNS (2005)Google Scholar
  14. 14.
    Wan, Y., Chen, G., Xu, Y.: A note on the minimum label spanning tree. IPL 84(2), 99–101 (2002)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    West, D.: Introduction to graph theory. Prentice Hall Inc., Upper Saddle River (1996)MATHGoogle Scholar
  16. 16.
    Wirth, H.: Multicriteria Approximation of Network Design and Network Upgrade Problems. PhD thesis, Department of Computer Science, Universität Würzburg, Germany (2005)Google Scholar
  17. 17.
    Xiong, Y.: The Minimum Labeling Spanning Tree Problem and Some Variants. PhD thesis, Graduate School of the University of Maryland, USA (2005)Google Scholar
  18. 18.
    Xiong, Y., Golden, B., Wasil, E.: A one-parameter genetic algorithm for the minimum labeling spanning tree problem. IEEE Transactions on Evolutionary Computation 9(1), 55–60 (2005)CrossRefGoogle Scholar
  19. 19.
    Xiong, Y., Golden, B., Wasil, E.: Worst case behavior of the MVCA heuristic for the minimum labeling spanning tree problem. Operations Research Letters 33(1), 77–80 (2005)CrossRefMATHGoogle Scholar
  20. 20.
    Zhang, P., Tang, L., Zhao, W., Cai, J., Li, A.: Approximation and hardness results for label cut and related problems (manuscript) (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Michael R. Fellows
    • 1
  • Jiong Guo
    • 2
  • Iyad A. Kanj
    • 3
  1. 1.The University of NewcastleCallaghanAustralia
  2. 2.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany
  3. 3.School of ComputingDePaul UniversityChicagoUSA

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