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Abstract

Treewidth is a graph parameter with several interesting theoretical and practical applications. This survey reviews algorithmic results on determining the treewidth of a given graph, and finding a tree decomposition of small width. Both theoretical results, establishing the asymptotic computational complexity of the problem, as experimental work on heuristics (both for upper bounds as for lower bounds), preprocessing, exact algorithms, and postprocessing are discussed.

Keywords

Input Graph Tree Decomposition Chordal Graph Reduction Rule Linear Time Algorithm 
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.
    Alber, J., Dorn, F., Niedermeier, R.: Experimental evaluation of a tree decomposition based algorithm for vertex cover on planar graphs. To appear in Discrete Applied Mathematics (2004)Google Scholar
  2. 2.
    Amir, E.: Efficient approximations for triangulation of minimum treewidth. In: Proc. 17th Conference on Uncertainty in Artificial Intelligence, pp. 7–15 (2001)Google Scholar
  3. 3.
    Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Alg. Disc. Meth. 8, 277–284 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Arnborg, S., Proskurowski, A.: Characterization and recognition of partial 3-trees. SIAM J. Alg. Disc. Meth. 7, 305–314 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bachoore, E., Bodlaender, H.L.: New upper bound heuristics for treewidth. Technical Report UU-CS-2004-036, Institute for Information and Computing Sciences, Utrecht University, Utrecht, the Netherlands (2004)Google Scholar
  6. 6.
    Becker, A., Geiger, D.: A sufficiently fast algorithm for finding close to optimal clique trees. Artificial Intelligence 125, 3–17 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Berry, A., Blair, J., Heggernes, P., Peyton, B.: Maximum cardinality search for computing minimal triangulations of graphs. Algorithmica 39, 287–298 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Blair, J.R.S., Heggernes, P., Telle, J.: A practical algorithm for making filled graphs minimal. Theor. Comp. Sc. 250, 125–141 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybernetica 11, 1–23 (1993)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Bodlaender, H.L.: Improved self-reduction algorithms for graphs with bounded treewidth. Disc. Appl. Math. 54, 101–115 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comp. Sc. 209, 1–45 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bodlaender, H.L., Gilbert, J.R., Hafsteinsson, H., Kloks, T.: Approximating treewidth, pathwidth, frontsize, and minimum elimination tree height. J. Algorithms 18, 238–255 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Bodlaender, H.L., Hagerup, T.: Parallel algorithms with optimal speedup for bounded treewidth. SIAM J. Comput. 27, 1725–1746 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21, 358–402 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Bodlaender, H.L., Kloks, T., Kratsch, D., Mueller, H.: Treewidth and minimum fill-in on d-trapezoid graphs. J. Graph Algorithms and Applications 2(5), 1–23 (1998)MathSciNetGoogle Scholar
  17. 17.
    Bodlaender, H.L., Koster, A.M.C.A.: On the maximum cardinality search lower bound for treewidth. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 81–92. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Bodlaender, H.L., Koster, A.M.C.A.: Safe separators for treewidth. In: Proceedings 6th Workshop on Algorithm Engineering and Experiments, ALENEX 2004, pp. 70–78 (2004)Google Scholar
  19. 19.
    Bodlaender, H.L., Koster, A.M.C.A., van den Eijkhof, F., van der Gaag, L.C.: Pre-processing for triangulation of probabilistic networks. In: Breese, J., Koller, D. (eds.) Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence, pp. 32–39. Morgan Kaufmann, San Francisco (2001)Google Scholar
  20. 20.
    Bodlaender, H.L., Koster, A.M.C.A., Wolle, T.: Contraction and treewidth lower bounds. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 628–639. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  21. 21.
    Bodlaender, H.L., Koster, A.M.C.A., Wolle, T.: Degree-based treewidth lower bounds (2004) (Paper in preparation)Google Scholar
  22. 22.
    Bodlaender, H.L., Möhring, R.H.: The pathwidth and treewidth of cographs. SIAM J. Disc. Math. 6, 181–188 (1993)zbMATHCrossRefGoogle Scholar
  23. 23.
    Bodlaender, H.L., Rotics, U.: Computing the treewidth and the minimum fill-in with the modular decomposition. Algorithmica 36, 375–408 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Bodlaender, H.L., Thilikos, D.M.: Treewidth for graphs with small chordality. Disc. Appl. Math. 79, 45–61 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Bodlaender, H.L., van Antwerpen-de Fluiter, B.: Parallel algorithms for series parallel graphs and graphs with treewidth two. Algorithmica 29, 543–559 (2001)Google Scholar
  26. 26.
    Bodlaender, H.L., Wolle, T.: Contraction degeneracy on cographs. Technical Report UU-CS-2004-031, Institute for Information and Computing Sciences, Utrecht University, Utrecht, the Netherlands (2004)Google Scholar
  27. 27.
    Bouchitté, V., Kratsch, D., Müller, H., Todinca, I.: On treewidth approximations. Disc. Appl. Math. 136, 183–196 (2004)zbMATHCrossRefGoogle Scholar
  28. 28.
    Bouchitté, V., Todinca, I.: Treewidth and minimum fill-in: Grouping the minimal separators. SIAM J. Comput. 31, 212–232 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Bouchitté, V., Todinca, I.: Listing all potential maximal cliques of a graph. Theor. Comp. Sc. 276, 17–32 (2002)zbMATHCrossRefGoogle Scholar
  30. 30.
    Bouchitté, V., Todinca, I.: Approximating the treewidth of at-free graphs. Disc. Appl. Math. 131, 11–37 (2003)zbMATHCrossRefGoogle Scholar
  31. 31.
    Broersma, H., Dahlhaus, E., Kloks, T.: A linear time algorithm for minimum fill in and tree width for distance hereditary graphs. Disc. Appl. Math. 99, 367–400 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Broersma, H., Kloks, T., Kratsch, D., Müller, H.: A generalization of AT-free graphs and a generic algorithm for solving triangulation problems. Algorithmica 32, 594–610 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Clautiaux, F., Moukrim, S.N.A., Carlier, J.: Heuristic and meta-heuristic methods for computing graph treewidth. RAIRO Oper. Res. 38, 13–26 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Clautiaux, F., Carlier, J., Moukrim, A., Négre, S.: New lower and upper bounds for graph treewidth. In: Jansen, K., Margraf, M., Mastrolli, M., Rolim, J.D.P. (eds.) WEA 2003. LNCS, vol. 2647, pp. 70–80. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  35. 35.
    Cook, W., Seymour, P.D.: Tour merging via branch-decomposition. Informs J. on Computing 15(3), 233–248 (2003)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique width. Theor. Comp. Sc. 33, 125–150 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Dahlhaus, E.: Minimal elimination ordering inside a given chordal graph. In: Möhring, R.H. (ed.) WG 1997. LNCS, vol. 1335, pp. 132–143. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  38. 38.
    Dahlhaus, E.: Minimum fill-in and treewidth for graphs modularly decomposable into chordal graphs. In: Hromkovič, J., Sýkora, O. (eds.) WG 1998. LNCS, vol. 1517, pp. 351–358. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  39. 39.
    Dechter, R.: Bucket elimination: a unifying framework for reasoning. Acta Informatica 113, 41–85 (1999)zbMATHMathSciNetGoogle Scholar
  40. 40.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  41. 41.
    Fomin, F.V., Kratsch, D., Todinca, I.: Exact (exponential) algorithms for treewidth and minimum fill-in. In: Proceedings of the 31st International Colloquium on Automata, Languages and Programming, pp. 568–580 (2004)Google Scholar
  42. 42.
    Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Comb. Theory Series B 16, 47–56 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Gogate, V., Dechter, R.: A complete anytime algorithm for treewidth. In: Proceedings UAI 2004, Uncertainty in Artificial Intelligence (2004)Google Scholar
  44. 44.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)zbMATHGoogle Scholar
  45. 45.
    Gustedt, J., Mæhle, O.A., Telle, J.A.: The treewidth of Java programs. In: Mount, D.M., Stein, C. (eds.) ALENEX 2002. LNCS, vol. 2409, pp. 86–97. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  46. 46.
    Habib, M., Möhring, R.H.: Treewidth of cocomparability graphs and a new order-theoretic parameter. ORDER 1, 47–60 (1994)CrossRefGoogle Scholar
  47. 47.
    Heggernes, P., Telle, J.A., Villanger, Y.: Computing minimal triangulations in time O(n αlogn) = o(n 2.376). To appear in proceedings SODA 2005 (2005)Google Scholar
  48. 48.
    Heggernes, P., Villanger, Y.: Efficient implementation of a minimal triangulation algorithm. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 550–561. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  49. 49.
    Held, M., Karp, R.: A dynamic programming approach to sequencing problems. J. SIAM 10, 196–210 (1962)zbMATHMathSciNetGoogle Scholar
  50. 50.
    Hicks, I.V.: Graphs, branchwidth, and tangles! Oh my! Working paper, http://ie.tamu.edu/People/faculty/Hicks/default.htm
  51. 51.
    Hicks, I.V.: Planar branch decompositions I: The ratcatcher. INFORMS Journal on Computing (to appear)Google Scholar
  52. 52.
    Hicks, I.V.: Planar branch decompositions II: The cycle method. INFORMS Journal on Computing (to appear)Google Scholar
  53. 53.
    Hicks, I.V.: Branch Decompositions and their Applications. Ph. d. thesis, Rice University, Houston, Texas (2000)Google Scholar
  54. 54.
    Hicks, I.V.: Branchwidth heuristics. Congressus Numerantium 159, 31–50 (2002)zbMATHMathSciNetGoogle Scholar
  55. 55.
    Hicks, I.V.: Branch decompositions and minor containment. Networks 43, 1–9 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Kjærulff, U.: Optimal decomposition of probabilistic networks by simulated annealing. Statistics and Computing 2, 2–17 (1992)CrossRefGoogle Scholar
  57. 57.
    Kloks, T.: Treewidth. Computations and Approximations. In: Kloks, T. (ed.) Treewidth. LNCS, vol. 842. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  58. 58.
    Kloks, T.: Treewidth of circle graphs. Int. J. Found. Computer Science 7, 111–120 (1996)zbMATHCrossRefGoogle Scholar
  59. 59.
    Kloks, T., Kratsch, D.: Treewidth of chordal bipartite graphs. J. Algorithms 19, 266–281 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  60. 60.
    Kloks, T., Kratsch, D., Spinrad, J.: On treewidth and minimum fill-in of asteroidal triple-free graphs. Theor. Comp. Sc. 175, 309–335 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  61. 61.
    Koster, A.M.C.A.: Frequency Assignment - Models and Algorithms. PhD thesis, Univ. Maastricht, Maastricht, the Netherlands (1999)Google Scholar
  62. 62.
    Koster, A.M.C.A., Bodlaender, H.L., van Hoesel, S.P.M.: Treewidth: Computational experiments. In: Broersma, H., Faigle, U., Hurink, J., Pickl, S. (eds.) Electronic Notes in Discrete Mathematics, vol. 8. Elsevier Science Publishers, Amsterdam (2001)Google Scholar
  63. 63.
    Koster, A.M.C.A., van Hoesel, S.P.M., Kolen, A.W.J.: Solving partial constraint satisfaction problems with tree decomposition. Networks 40, 170–180 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    Lagergren, J.: Efficient parallel algorithms for graphs of bounded tree-width. J. Algorithms 20, 20–44 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    Lagergren, J., Arnborg, S.: Finding minimal forbidden minors using a finite congruence. In: Leach Albert, J., Monien, B., Rodríguez-Artalejo, M. (eds.) ICALP 1991. LNCS, vol. 510, pp. 532–543. Springer, Heidelberg (1991)Google Scholar
  66. 66.
    Larrañaga, P., Kuijpers, C.M.H., Poza, M., Murga, R.H.: Decomposing Bayesian networks: triangulation of the moral graph with genetic algorithms. Statistics and Computing (UK) 7(1), 19–34 (1997)CrossRefGoogle Scholar
  67. 67.
    Lauritzen, S.J., Spiegelhalter, D.J.: Local computations with probabilities on graphical structures and their application to expert systems. The Journal of the Royal Statistical Society. Series B (Methodological) 50, 157–224 (1988)zbMATHMathSciNetGoogle Scholar
  68. 68.
    Lick, D.R., White, A.T.: k-degenerate graphs. Canadian Journal of Mathematics 22, 1082–1096 (1970)zbMATHMathSciNetCrossRefGoogle Scholar
  69. 69.
    Lucena, B.: A new lower bound for tree-width using maximum cardinality search. SIAM J. Disc. Math. 16, 345–353 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  70. 70.
    Matoušek, J., Thomas, R.: Algorithms for finding tree-decompositions of graphs. J. Algorithms 12, 1–22 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    Olesen, K.G., Madsen, A.L.: Maximal prime subgraph decomposition of Bayesian networks. IEEE Trans. on Systems, Man, and Cybernetics, Part B 32, 21–31 (2002)CrossRefGoogle Scholar
  72. 72.
    Parra, A., Scheffler, P.: Characterizations and algorithmic applications of chordal graph embeddings. Disc. Appl. Math. 79, 171–188 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  73. 73.
    Perković, L., Reed, B.: An improved algorithm for finding tree decompositions of small width. In: Widmayer, P., Neyer, G., Eidenbenz, S. (eds.) WG 1999. LNCS, vol. 1665, pp. 148–154. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  74. 74.
    Ramachandramurthi, S.: A lower bound for treewidth and its consequences. In: Mayr, E.W., Schmidt, G., Tinhofer, G. (eds.) WG 1994. LNCS, vol. 903, pp. 14–25. Springer, Heidelberg (1995)Google Scholar
  75. 75.
    Ramachandramurthi, S.: The structure and number of obstructions to treewidth. SIAM J. Disc. Math. 10, 146–157 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  76. 76.
    Reed, B.: Finding approximate separators and computing tree-width quickly. In: Proceedings of the 24th Annual Symposium on Theory of Computing, pp. 221–228. ACM Press, New York (1992)Google Scholar
  77. 77.
    Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7, 309–322 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  78. 78.
    Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory Series B 63, 65–110 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  79. 79.
    Röhrig, H.: Tree decomposition: A feasibility study. Master’s thesis, Max-Planck-Institut für Informatik, Saarbrücken, Germany (1998)Google Scholar
  80. 80.
    Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5, 266–283 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  81. 81.
    Sanders, D.P.: On linear recognition of tree-width at most four. SIAM J. Disc. Math. 9(1), 101–117 (1996)zbMATHCrossRefGoogle Scholar
  82. 82.
    Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14(2), 217–241 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  83. 83.
    Shoikhet, K., Geiger, D.: A practical algorithm for finding optimal triangulations. In: Proc. National Conference on Artificial Intelligence (AAAI 1997), pp. 185–190. Morgan Kaufmann, San Francisco (1997)Google Scholar
  84. 84.
    Szekeres, G., Wilf, H.S.: An inequality for the chromatic number of a graph. J. Comb. Theory 4, 1–3 (1968)CrossRefMathSciNetGoogle Scholar
  85. 85.
    Tarjan, R.E., Yannakakis, M.: Simple linear time algorithms to test chordiality of graphs, test acyclicity of graphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13, 566–579 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  86. 86.
    Thorup, M.: Structured programs have small tree-width and good register allocation. Information and Computation 142, 159–181 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  87. 87.
  88. 88.
    Valdes, J., Tarjan, R.E., Lawler, E.L.: The recognition of series parallel digraphs. SIAM J. Comput. 11, 298–313 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  89. 89.
    van den Eijkhof, F., Bodlaender, H.L.: Safe reduction rules for weighted treewidth. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 176–185. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  90. 90.
    Woeginger, G.J.: Exact algorithms for NP-hard problems: A survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  91. 91.
    Yamaguchi, A., Aoki, K.F., Mamitsuka, H.: Graph complexity of chemical compounds in biological pathways. Genome Informatics 14, 376–377 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Hans L. Bodlaender
    • 1
  1. 1.Institute of Information and Computing SciencesUtrecht UniversityUtrechtthe Netherlands

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