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Theory of Computing Systems

, Volume 55, Issue 2, pp 330–346 | Cite as

An Extended Tree-Width Notion for Directed Graphs Related to the Computation of Permanents

  • Klaus MeerEmail author
Article
  • 123 Downloads

Abstract

It is well known that permanents of matrices of bounded tree-width are efficiently computable. Here, the tree-width of a square matrix M=(m ij ) with entries from a field \(\mathbb{K}\) is the tree-width of the underlying graph G M having an edge (i,j) if and only if the entry m ij ≠0. Though G M is directed this does not influence the tree-width definition. Thus, it does not reflect the lacking symmetry when m ij ≠0 but m ji =0. The latter however might have impact on the computation of the permanent.

In this paper we introduce and study an extended notion of tree-width for directed graphs called triangular tree-width. We give examples where the latter parameter is bounded whereas the former is not. As main result we show that permanents of matrices of bounded triangular tree-width are efficiently computable. This result is shown to hold as well for the Hamiltonian Cycle problem.

Keywords

Treewidth notions for digraphs Efficient algorithms Computation of the permanent 

Notes

Acknowledgements

Thanks are due to the anonymous referees for helpful comments improving readability of the paper and for pointing out some open problems.

References

  1. 1.
    Arnborg, S., Corneil, D., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Matrix Anal. Appl. 8(2), 277–284 (1987) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Barvinok, A.: Two algorithmic results for the traveling salesman problem. Math. Oper. Res. 21, 65–84 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Berwanger, D., Dawar, A., Hunter, P., Kreutzer, S.: DAG-width and parity games. In: Proceedings STACS’06. Lecture Notes in Computer Science, vol. 3884, pp. 524–536 (2006) Google Scholar
  4. 4.
    Berwanger, D., Grädel, E.: Entanglement—a measure for the complexity of directed graphs with applications to logic and games. In: Proc. LPAR 2004. LNCS, vol. 3452, pp. 209–223. Springer, Berlin (2005) Google Scholar
  5. 5.
    Briquel, I., Koiran, P., Meer, K.: On the expressive power of CNF formulas of bounded tree- and clique-width. Discrete Appl. Math. 159(1), 1–14 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory. Algorithms and Computation in Mathematics, vol. 7. Springer, Berlin (2000) zbMATHGoogle Scholar
  7. 7.
    Courcelle, B.: The monadic second order logic for graphs I: recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Courcelle, B., Makowsky, J.A., Rotics, U.: On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discrete Appl. Math. 108(1–2), 23–52 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Fischer, E., Makowsky, J., Ravve, E.V.: Counting truth assignments of formulas of bounded tree-width or clique-width. Discrete Appl. Math. 156, 511–529 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Flarup, U., Koiran, P., Lyaudet, L.: On the expressive power of planar perfect matching and permanents of bounded tree-width matrices. In: Proc. 18th International Symposium ISAAC. Lecture Notes in Computer Science, vol. 4835, pp. 124–136. Springer, Berlin (2007) Google Scholar
  11. 11.
    Ganian, R., Hlinĕný, P., Kneis, J., Meister, D., Obdrz̆álek, J., Rossmanith, P., Sikdar, S.: Are there any good digraph width measures? In: Proceedings of the International Symposium on Parameterized and Exact Computation IPEC 2010. Lecture Notes in Computer Science, vol. 6478, pp. 135–146. Springer, Berlin (2010) Google Scholar
  12. 12.
    Hunter, P., Kreutzer, S.: Digraph measures: Kelly decompositions, games, and orderings. Theor. Comput. Sci. 399(3), 206–219 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. J. Comb. Theory, Ser. B 82, 138–154 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Kloks, T.: Treewidth, Computations and Approximations. Lecture Notes in Computer Science, vol. 842. Springer, Berlin (1994) zbMATHGoogle Scholar
  15. 15.
    Obdrz̆álek, J.: DAG-width—connectivity measure for directed graphs. In: Proc. SODA’06, ACM-SIAM, pp. 814–821 (2006) Google Scholar
  16. 16.
    Thomassen, C.: Embeddings and minors. In: Grötschel, M., Lovász, L., Graham, R.L. (eds.) Handbook of Combinatoris, pp. 302–349. North-Holland, Amsterdam (1995). Chapter 5 Google Scholar
  17. 17.
    Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8(2), 189–201 (1979) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Valiant, L.G.: Completeness classes in algebra. In: Proc. 11th ACM Symposium on Theory of Computing 1979, pp. 249–261. ACM, New York (1979) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Lehrstuhl Theoretische InformatikBTU CottbusCottbusGermany

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