On Listing, Sampling, and Counting the Chordal Graphs with Edge Constraints

  • Shuji Kijima
  • Masashi Kiyomi
  • Yoshio Okamoto
  • Takeaki Uno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5092)


We discuss the problems to list, sample, and count the chordal graphs with edge constraints. The objects we look at are chordal graphs sandwiched by a given pair of graphs where we assume at least one of the input pair is chordal. The setting is a natural generalization of chordal completions and deletions. For the listing problem, we give an efficient algorithm running in polynomial time per output with polynomial space. As for the sampling problem, we give two clues that seem to imply that a random sampling is not easy. The first clue is that we show #P-completeness results for counting problems. The second clue is that we give an instance for which a natural Markov chain suffers from an exponential mixing time. These results provide a unified viewpoint from algorithms theory to problems arising from various areas such as statistics, data mining, and numerical computation.


Markov Chain SIAM Journal Interval Graph Chordal Graph Listing 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Avis, D., Fukuda, K.: Reverse Search for Enumeration. Discrete Applied Mathematics 65, 21–46 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Dirac, G.A.: On Rigid Circuit Graphs. Abhandl. Math. Seminar Univ. Hamburg 25, 71–76 (1961)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Fulkerson, D.R., Gross, O.A.: Incidence Matrices and Interval Graphs. Pacific Journal of Mathematics 15, 835–855 (1965)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Fomin, F.V., Kratsch, D., Todinca, I.: Exact (Exponential) Algorithms for Treewidth and Minimum Fill-in. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 568–580. Springer, Heidelberg (2004)Google Scholar
  5. 5.
    Fukuda, M., Kojima, M., Murota, K., Nakata, K.: Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework. SIAM Journal on Optimization 11, 647–674 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)zbMATHGoogle Scholar
  7. 7.
    Golumbic, M.C., Kaplan, H., Shamir, R.: Graph Sandwich Problems. Journal of Algorithms 19, 449–473 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Heggernes, P.: Minimal Triangulations of Graphs: A Survey. Discrete Mathematics 306, 297–317 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Heggernes, P., Suchan, K., Todinca, I., Villanger, Y.: Characterizing Minimal Interval Vompletions: towards Better Understanding of Profile and Pathwidth. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 236–247. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Ibarra, L.: Fully Dynamic Algorithms for Chordal Graphs. In: Proc. of SODA 1999, pp. 923–924 (1999)Google Scholar
  11. 11.
    Kijima, S., Kiyomi, M., Okamoto, Y., Uno, T.: On Counting, Sampling, and Listing of Chordal Graphs with Edge Constrains. RIMS-1610, Kyoto University (preprint, 2007),
  12. 12.
    Kiyomi, M., Uno, T.: Generating Chordal Graphs Included in Given Graphs. IEICE Transactions on Information and Systems E89-D, 763–770 (2006)CrossRefGoogle Scholar
  13. 13.
    Kiyomi, M., Kijima, S., Uno, T.: Listing Chordal Graphs and Interval Graphs. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 68–77. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Kloks, T., Bodlaender, H.L., Müller, H., Kratsch, D.: Computing Treewidth and Minimum Fill-in: All You Need are the Minimal Separators. In: Lengauer, T. (ed.) ESA 1993. LNCS, vol. 726, pp. 260–271. Springer, Heidelberg (1993)Google Scholar
  15. 15.
    Kloks, T., Bodlaender, H.L., Müller, H., Kratsch, D.: Erratum to the ESA 1993 proceedings. In: van Leeuwen, J. (ed.) ESA 1994. LNCS, vol. 855, p. 508. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  16. 16.
    Leckerkerker, C.G., Boland, J.C.: Representation of a Finite Graph by a Set of Intervals on the Real Line. Fundamenta Mathematicae 51, 45–64 (1962)MathSciNetGoogle Scholar
  17. 17.
    Marx, D.: Chordal Deletion is Fixed-Parameter Tractable. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 37–48. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    Natanzon, A., Shamir, R., Sharan, R.: A Polynomial Approximation Algorithm for the Minimum Fill-in Problem. SIAM Journal on Computing 30, 1067–1079 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Natanzon, A., Shamir, R., Sharan, R.: Complexity Classification of Some Edge Modification Problems. Discrete Applied Mathematics 113, 109–128 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Pedersen, T., Bruce, R.F., Wiebe, J.: Sequential Model Selection for Word Sense Disambiguation. In: Proceedings of the Fifth Conference on Applied Natural Language Processing (ANLP 1997), pp. 388–395 (1997)Google Scholar
  21. 21.
    Robertson, N., Seymour, P.: Graph Minors II. Algorithmic Aspects of Tree-Width. Journal of Algorithms 7, 309–322 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Rose, D.J.: A Graph-Theoretic Study of the Numerical Solution of Sparse Positive Definite Systems of Linear Equations. In: Read, R.C. (ed.) Graph Theory and Computing, pp. 183–217. Academic Press, New York (1972)Google Scholar
  23. 23.
    Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic Aspects of Vertex Elimination on Graphs. SIAM Journal on Computing 5, 266–283 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Sinclair, A.: Algorithms for Random Generation and Counting: A Markov Chain Approach. Birkhäuser, Boston (1993)zbMATHGoogle Scholar
  25. 25.
    Tarjan, R.E., Yannakakis, M.: Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs. SIAM Journal on Computing 13, 566–579 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Takemura, A., Endo, Y.: Evaluation of Per-record Identification Risk and Swappability of Records in a Microdata Set via Decomposable Models. arXiv:math.ST/0603609Google Scholar
  27. 27.
    Valiant, V.G.: The Complexity of Computing the Permanent. Theoretical Computer Science 8, 189–201 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Valiant, V.G.: The Complexity of Enumeration and Reliability Problems. SIAM Journal on Computing 8, 410–421 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Yamashita, N.: Sparse Quasi-Newton Updates with Positive Definite Matrix Completion. Mathematical Programming (2007),
  30. 30.
    Yannakakis, M.: Computing the Minimum Fill-in is NP-Complete. SIAM Journal on Algebraic and Discrete Methods 2, 77–79 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Whittaker, J.: Graphical Models in Applied Multivariate Statistics. Wiley, New York (1990)zbMATHGoogle Scholar
  32. 32.
    Wormald, N.C.: Counting Labeled Chordal Graphs. Graphs and Combinatorics 1, 193–200 (1985)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Shuji Kijima
    • 1
  • Masashi Kiyomi
    • 2
  • Yoshio Okamoto
    • 3
  • Takeaki Uno
    • 4
  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  2. 2.School of Information ScienceJapan Advanced Institute of Science and TechnologyNomiJapan
  3. 3.Graduate School of Information Science and EngineeringTokyo Institute of TechnologyTokyoJapan
  4. 4.National Institute of InformaticsTokyoJapan

Personalised recommendations