The Linear Arrangement Problem Parameterized Above Guaranteed Value

  • Gregory Gutin
  • Arash Rafiey
  • Stefan Szeider
  • Anders Yeo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3998)

Abstract

A linear arrangement (LA) is an assignment of distinct integers to the vertices of a graph. The cost of an LA is the sum of lengths of the edges of the graph, where the length of an edge is defined as the absolute value of the difference of the integers assigned to its ends. For many application one hopes to find an LA with small cost. However, it is a classical NP-complete problem to decide whether a given graph G admits an LA of cost bounded by a given integer. Since every edge of G contributes at least one to the cost of any LA, the problem becomes trivially fixed-parameter tractable (FPT) if parameterized by the upper bound of the cost. Fernau asked whether the problem remains FPT if parameterized by the upper bound of the cost minus the number of edges of the given graph; thus whether the problem is FPT “parameterized above guaranteed value.” We answer this question positively by deriving an algorithm which decides in time O(m + n + 5.88k) whether a given graph with m edges and n vertices admits an LA of cost at most m + k (the algorithm computes such an LA if it exists). Our algorithm is based on a procedure which generates a problem kernel of linear size in linear time for a connected graph G. We also prove that more general parameterized LA problems stated by Serna and Thilikos are not FPT, unless P = NP.

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References

  1. 1.
    Cesati, M.: Compendium of parameterized problems (September 2005), http://bravo.ce.uniroma2.it/home/cesati/research/compendium.pdf
  2. 2.
    Chung, F.R.K.: On optimal linear arrangements of trees. Comp. & Maths. with Appls. 10, 43–60 (1984)MATHCrossRefGoogle Scholar
  3. 3.
    Diestel, R.: Graph Theory, 2nd edn. Springer, New York (2000)Google Scholar
  4. 4.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)Google Scholar
  5. 5.
    Estivill-Castro, V., Fellows, M.R., Langston, M.A., Rosamond, F.A.: FPT is P-Time extremal structure I. In: Broersma, H., Johnson, M., Szeider, S. (eds.) Algorithms and Complexity in Durham 2005, Proceedings of the first ACiD Workshop. Texts in Algorithmics, vol. 4, pp. 1–41. King’s College Publications (2005)Google Scholar
  6. 6.
    Fernau, H.: Parameterized Algorithmics: A Graph-theoretic Approach. Habilitation thesis, U. Tübingen (2005)Google Scholar
  7. 7.
    Fernau, H.: Parameterized Algorithmics for Linear Arrangement Problems. Talk at Dagstuhl (July 2005), slides at: http://www.dagstuhl.de/files/Materials/05/05301/05301.FernauHenning.Slides.pdf
  8. 8.
    Fernau, H.: Parameterized Algorithmics for Linear Arrangement Problems (manscript) (July 2005), http://homepages.feis.herts.ac.uk/~comrhf/papers/ola.pdf
  9. 9.
    Flum, J., Grohe, M.: Describing parameterized complexity classes. Information and Computation 187, 291–319 (2003)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)Google Scholar
  11. 11.
    Garey, M.R., Johnson, D.R.: Computers and Intractability. W.H. Freeman & Comp., New York (1979)MATHGoogle Scholar
  12. 12.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theoret. Comput. Sci. 1, 237–267 (1976)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Goldberg, M.K., Klipker, I.A.: Minimal placing pf trees on a line. Tech. Report, Physico-Technical Institute of Low Temperatures, Ukranian SSR Acad. of Sciences, USSR (1976) (in Russian)Google Scholar
  14. 14.
    Harper, L.H.: Optimal assignments of numbers to vertices. J. Soc. Indust. Appl. Math. 12, 131–135 (1964)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. J. Algorithms 31, 335–354 (1999)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (forthcoming, 2006)Google Scholar
  17. 17.
    Shiloach, Y.: A minimum linear arrangement algorithm for undirected trees. SIAM J. Comp. 8, 15–32 (1979)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Serna, M., Thilikos, D.M.: Parameterized complexity for graph layout problems. EATCS Bulletin 86, 41–65 (2005)MATHMathSciNetGoogle Scholar
  19. 19.
    Tarjan, R.E.: Depth first search and linear graph algorithms. SIAM J. Comput. 1, 146–160 (1972)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Gregory Gutin
    • 1
  • Arash Rafiey
    • 1
  • Stefan Szeider
    • 2
  • Anders Yeo
    • 1
  1. 1.Department of Computer ScienceRoyal Holloway University of LondonEgham, SurreyEngland, United Kingdom
  2. 2.Department of Computer ScienceDurham UniversityDurhamEngland, United Kingdom

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