On the Algorithmic Effectiveness of Digraph Decompositions and Complexity Measures

  • Michael Lampis
  • Georgia Kaouri
  • Valia Mitsou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)

Abstract

We place our focus on the gap between treewidth’s success in producing fixed-parameter polynomial algorithms for hard graph problems, and specifically Hamiltonian Circuit and Max Cut, and the failure of its directed variants (directed tree-width [9], DAG-width [11] and kelly-width [8]) to replicate it in the realm of digraphs. We answer the question of why this gap exists by giving two hardness results: we show that Directed Hamiltonian Circuit is W[2]-hard when the parameter is the width of the input graph, for any of these widths, and that Max Di Cut remains NP-hard even when restricted to DAGs, which have the minimum possible width under all these definitions. Our results also apply to directed pathwidth.

Keywords

Treewidth Digraph decompositions Parameterized Complexity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barát, J.: Directed path-width and monotonicity in digraph searching. Graphs and Combinatorics 22(2), 161–172 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bodlaender, H.L.: Treewidth: Algorithmic techniques and results. In: Privara, I., Ružička, P. (eds.) MFCS 1997. LNCS, vol. 1295, pp. 19–36. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L.: Treewidth: Characterizations, applications, and computations. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 1–14. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L.: Treewidth: Structure and algorithms. In: Prencipe, G., Zaks, S. (eds.) SIROCCO 2007. LNCS, vol. 4474, pp. 11–25. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Courcelle, B.: The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs. Inf. Comput. 85(1), 12–75 (1990)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefMATHGoogle Scholar
  7. 7.
    Flum, J., Grohe, M.: Parameterized Complexity Theory, 1st edn. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2006)MATHGoogle Scholar
  8. 8.
    Hunter, P., Kreutzer, S.: Digraph measures: Kelly decompositions, games, and orderings. In: Bansal, N., Pruhs, K., Stein, C. (eds.) SODA, pp. 637–644. SIAM, Philadelphia (2007)Google Scholar
  9. 9.
    Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. J. Comb. Theory, Ser. B 82(1), 138–154 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, USA (2006)CrossRefMATHGoogle Scholar
  11. 11.
    Obdrzálek, J.: Dag-width: connectivity measure for directed graphs. In: SODA, pp. 814–821. ACM Press, New York (2006)CrossRefGoogle Scholar
  12. 12.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43(3), 425–440 (1991)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic Aspects of Tree-Width. J. Algorithms 7(3), 309–322 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Michael Lampis
    • 1
  • Georgia Kaouri
    • 2
  • Valia Mitsou
    • 1
  1. 1.City University of New YorkUSA
  2. 2.National Technical University of AthensGreece

Personalised recommendations