Cybernetics and Systems Analysis

, Volume 43, Issue 4, pp 549–562 | Cite as

Tree decomposition and discrete optimization problems: A survey

  • O. A. Shcherbina
Article

Abstract

The paper considers tree decomposition methods as applied to discrete optimization and presents relevant mathematical results.

Keywords

tree decomposition discrete optimization graphs trees triangulation 

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References

  1. 1.
    V. S. Mikhalevich, I. V. Sergienko, A. I. Kuksa, V. A. Roshchin, and V. A. Trubin, Results of Experimental Study of the Efficiency of Methods included into the DISPRO Software [in Russian], Prepr. AN USSR, IK, 80-47, Kyiv (1980).Google Scholar
  2. 2.
    Yu. I. Zhuravlyov, Selected Scientific Works [in Russian], Magistr, Moscow (1998).Google Scholar
  3. 3.
    O. A. Shcherbina, “Application of local algorithms to solving integer linear programming problems,” in: Problems of Cybernetics [in Russian], NSK AN SSSR, Moscow (1989), pp. 19–34.Google Scholar
  4. 4.
    O. A. Shcherbina, “Nonserial modification of a local decomposition algorithm for discrete optimization problems,” in: Dynamic Systems [in Russian], Issue 19, TNU, Simferopol (2005), pp. 179–190.Google Scholar
  5. 5.
    B. Courcelle, “The monadic second-order logic of graphs I: Recognizable sets of finite graphs,” Inform. and Comput., 85, 12–75 (1990).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    S. Arnborg, D. G. Corneil, and A. Proskurowski, “Complexity of finding embeddings in a k-tree,” SIAM J. Alg. Disc. Math., 8, 277–284 (1987).MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    W. Cook and P. D. Seymour, “An algorithm for the ring-routing problem,” in: Bellcore Techn. Memorandum, Bellcore (1994).Google Scholar
  8. 8.
    W. Cook and P. D. Seymour, “Tour merging via branch decomposition,” Inform. J. Comput., 15, No. 3, 233–248 (2003).CrossRefMathSciNetGoogle Scholar
  9. 9.
    A. M. C. A. Koster, H. L. Bodlaender, and S. P. M. van Hoesel, “Treewidth: Computational experiments,” in: Zib-Report-38, Berlin (2001).Google Scholar
  10. 10.
    M. C. A. Koster, S. P. M van Hoesel, and A. W. J. Kolen, “Solving partial constraint satisfaction problems with tree decomposition,” Networks, 40, 170–180 (2002).MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    V. A. Emelichev, O. I. Mel’nikov, V. I. Sarvanov, and R. I. Tyshkevich, Lectures on Graph Theory [in Russian], Nauka, Moscow (1990).MATHGoogle Scholar
  12. 12.
    V. A. Evstigneev and V. N. Kasyanov, The Explanatory Dictionary on Graph Theory in Computer Science and Programming [in Russian], Nauka, Novosibirsk (1999) (http://pco.iis.nsk.su/grapp).Google Scholar
  13. 13.
    F. Harary, Graph Theory, Addison-Wesley, Reading, MA (1969).Google Scholar
  14. 14.
    N. Robertson and P. Seymour, “Graph minors II. Algorithmic aspects of treewidth,” J. Algorithms, 7, 309–322 (1986).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    H. L. Bodlaender, “Discovering treewidth,” in: Proc. SOFSEM Springer-Verlag, LNCS, 3381 (2006), pp. 1–16.MathSciNetGoogle Scholar
  16. 16.
    H. L. Bodlaender, “Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees,” J. Algorithms, 11, 631–643 (1990).MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    S. J. Lauritzen and D. J. Spiegelhalter, “Local computations with probabilities on graphical structures and their application to expert systems,” J. Roy. Statist. Soc., Ser. B, 50, 157–224 (1988).MATHMathSciNetGoogle Scholar
  18. 18.
    J. Alber, F. Dorn, and R. Niedermeier, “Experimental evaluation of a tree decomposition-based algorithm for vertex cover on planar graphs,” Disc. Appl. Math., 145, 210–219 (2004).MathSciNetGoogle Scholar
  19. 19.
    M. Commandeur, Solving Vertex Coloring using Tree Decompositions: Master’s Thesis, Univ. Maastricht (2004).Google Scholar
  20. 20.
    F. Gavril, “The intersection graphs of subtrees in trees are exactly the chordal graphs,” J. Combin. Theory, Ser. B, 16, 47–56 (1974).MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    J. A. George and J. W. H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall Inc., Englewood Cliffs (1981).MATHGoogle Scholar
  22. 22.
    S. Parter, “The use of linear graphs in Gauss elimination,” SIAM Review, 3, 119–130 (1961).MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    D. J. Rose, “A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations,” in: R. C. Read (ed.), Graph Theory and Computing, Acad. Press, New York (1972), pp. 183–217.Google Scholar
  24. 24.
    A. Hajnal and J. Surányi, “Über die Ausflösung von Graphen in vollständige Teilgraphen,” Ann. Univ. Sci. Budapest, No. 1, 113–121 (1958).Google Scholar
  25. 25.
    C. Beeri, R. Fagin, D. Maier, and M. Yannakakis, “On the desirability of acyclic database systems,” J. ACM, 30, 479–513 (1983).MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    R. E. Tarjan and M. Yannakakis, “Simple linear-time algorithms to test chordality of graphs, test acyclity of hypergraphs, and selectively reduce acyclic hypergraphs,” SIAM J. Comput., 13, 566–579 (1984).MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    S. L. Lauritzen and D. J. Spiegelhalter, “Local computations with probabilities on graphical structures and their application to expert systems,” J. Roy. Statist. Soc., 50, No. 2, 157–224 (1988).MATHMathSciNetGoogle Scholar
  28. 28.
    H. L. Bodlaender, “A tourist guide through treewidth,” Acta Cybernetica, 11, 1–21 (1993).MATHMathSciNetGoogle Scholar
  29. 29.
    M. Yannakakis, “Computing the minimum fill-in is NP-complete,” SIAM J. Alg. Disc. Math., 2, 77–79 (1981).MATHMathSciNetGoogle Scholar
  30. 30.
    C. E. Shannon, “The zero error capacity of a noisy channel,” IEEE Trans. Inform. Theory, 2, S8–S19 (1956).CrossRefMathSciNetGoogle Scholar
  31. 31.
    C. Berge “Färbung von Graphen, deren sämtliche bzw deren ungerade Kreise starr sind (Zusammenfassung),” Wiss. Z. Martin-Luther-Univ., Halle-Wittenberg, Math.-Natur. Reihe, 10, 114–115 (1961).Google Scholar
  32. 32.
    U. Kjaerulff, “A note on triangulated graphs and junction trees,” in: Lecture Note, Aalborg Univ. (1992).Google Scholar
  33. 33.
    M. Groetschel, L. Lovasz, and A. Schrijver, “Polynomial algorithms for perfect graphs,” Ann. Discrete Math., 21, 325–356 (1984).Google Scholar
  34. 34.
    M. Groetschel, L. Lovasz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, New York (1988).MATHGoogle Scholar
  35. 35.
    M. Gröetschel, L. Lovasz, and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization,” Combinatorica, 1, 169–197 (1981).CrossRefMathSciNetGoogle Scholar
  36. 36.
    T. Emden-Weinert, S. Hougardy, B. Kreuter, H. J. Prömel, and A. Steger, Einfuhrung in Graphen und Algorithmen (http://www.informatik.hu-berlin.de/alkox/lehre/skripte/ga).
  37. 37.
    A. Berry, “Graph extremities and minimal separation” (http://citeseer.ist.psu.edu/berry03graph.html).
  38. 38.
    G. A. Dirac, “On rigid circuit graphs,” Anh. Math. Sem. Univ. Hamburg, 25, 71–76 (1961).MATHMathSciNetGoogle Scholar
  39. 39.
    T. G. Lewis, B. W. Peyton, and A. Pothen, “A fast algorithm for reordering sparse matrices for parallel factorization,” SIAM J. Sci. Stat. Comput., 10, 1146–1173 (1989).MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    D. R. Fulkerson and O. A. Gross, “Incidence matrices and interval graphs,” Pacif. J. Math., 15, 835–855 (1965).MATHMathSciNetGoogle Scholar
  41. 41.
    D. Rose, R. E. Tarjan, and G. Lueker, “Algorithmic aspects of vertex elimination on graphs,” SIAM J. Comput., 5, 266–283 (1976).MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    E. Dahlhaus, P. L. Hammer, F. Maffray, and S. Olariu, “On domination elimination orderings and domination graphs,” in: Proc. 20th Intern. Workshop WG 94 (Graph-Theoret. Concepts in Comp. Sci.), LNCS, Springer Verlag, (1994), pp. 81–92.Google Scholar
  43. 43.
    D. G. Corneil, S. Olariu, and L. Stewart, “A linear time algorithm to compute a dominating path in AT-free graphs,” Inf. Proc. Letters, 54, 253–257 (1995).MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    A. Bretscher, D. G. Corneil, M. Habib, and C. Paul, “A simple linear time LexBFS cograph recognition algorithm-extended abstract,” in: Proc. 29th WG Workshop, LNCS, Springer-Verlag, 2880 (2003), pp. 119–130.Google Scholar
  45. 45.
    A. Berry, J. Blair, P. Heggernes, and B. Peyton, “Maximum cardinality search for computing minimal triangulations of graphs,” Algorithmica, 39, 287–298 (2004).MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    F. Gavril, “Algorithms for minimum coloring, maximum clique, minimum covering by cliques and maximum independent set of a chordal graph,” SIAM J. Comput., 1, 180–187 (1972).MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    J. R. Walter, “Representations of chordal graphs as subtrees of a tree,” J. Graph Theory, 2, 265–267 (1978).MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    P. Buneman, “A characterization of rigid circuit graphs,” Discrete Math., 9, 205–212 (1974).MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    P. A. Bernstein and N. Goodman, “Power of natural semijoins,” SIAM J. Comput., 10, No. 4, 751–771 (1981).MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Y. Shibata, “On the tree representation of chordal graphs,” J. Graph Theory, 12, 421–428 (1988).MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    C. W. Ho and R. C. T. Lee, “Counting clique trees and computing perfect elimination schemes in parallel,” Inform. Process. Lett., 31, No. 2, 61–68 (1989).MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    J. R. S. Blair, R. E. England, and M. G. Thomason, “Cliques and their separators in triangulated graphs,” Report CS-73-88, Dept. Computer Sci., Univ. Tennessee (1988).Google Scholar
  53. 53.
    J. R. S. Blair and B. W. Peyton, “An introduction to chordal graphs and clique trees,” in: J. A. George, J. R. Gilbert, and J. W. H. Liu (eds.), Sparse Matrix Computations: Graph Theory Issues and Algorithms, Springer Verlag, New York (1993), pp. 1–29.Google Scholar
  54. 54.
    Y. Villanger, “Efficient minimal triangulation of graphs by using tree decomposition,” PhD Thesis., Univ. of Bergen (2002).Google Scholar
  55. 55.
    G. Hajos, “Über eine Art von Graphen,” Internat. Math. Nachr., 47, Problem 65 (1957).Google Scholar
  56. 56.
    F. S. Roberts, Graph Theory and its Applications to Problems of Society, SIAM, Philadelphia, PA (1978).Google Scholar
  57. 57.
    M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Acad. Press, New York (1980).MATHGoogle Scholar
  58. 58.
    N. Deo, M. S. Krishnamoorthy, and M. A. Langston, “Exact and approximate solutions for the gate matrix layout problem,” IEEE T-CAD, 6, No. 1, 79–84 (1987).Google Scholar
  59. 59.
    P. C. Gilmore and A. J. Hoffman, “A characterization of comparability graphs and of interval graphs,” Can. J. Math., 16, 539–548 (1964).MATHMathSciNetGoogle Scholar
  60. 60.
    K. Menger, “Zur allgemeinen Kurventheorie,” Fund. Math., 10, 96–115 (1927).MATHGoogle Scholar
  61. 61.
    C. G. Lekkerkerker and J. C. Boland, “Representation of a finite graph by a set of intervals on the real line,” Fund. Math., 51, 45–64 (1962).MATHMathSciNetGoogle Scholar
  62. 62.
    A. Berry, P. Heggernes, and Y. Villanger, “A vertex incremental approach for maintaining chordality,” Disc. Math., 306, No. 3, 318–336 (2006) (http://www.ii.uib.no/∼pinar/incremental.pdf).MATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    D. Rose, “On simple characterizations of k-trees,” Disc. Math., 7, 317–322 (1974).MATHGoogle Scholar
  64. 64.
    P. Scheffler, Linear-Time Algorithms for NP-Complete Problems Restricted to Partial k-Trees, Report R-MATH 03/87, Prepr., Akad. der Wissenschaften der DDR, Karl Weierstrass Institut für Mathematik (1987).Google Scholar
  65. 65.
    T. V. Wimer, “Linear algorithms on k-terminal graphs,” PhD Thesis, Clemson Univ., Dept. of Computer Sci. (1987).Google Scholar
  66. 66.
    G. A. Dirac, “In abstrakten Graphen vorhandene vollständige 4-Graphen und ihre Unterteilungen,” Math. Nachr., 22, 61–85 (1960).MATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    J. A. Wald and C. J. Colbourn, “Steiner trees, partial 2-trees, and minimum IFI networks,” Networks, 13, 159–167 (1983).MATHCrossRefMathSciNetGoogle Scholar
  68. 68.
    N. Robertson and P. Seymour, “Graph minor III. Planar tree-width,” J. Combin. Theory (B), 36, 49–63 (1984).MATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    B. Courcelle, “Graph rewriting: An algebraic and logic approach,” in: J. Van Leeuwen (ed.), Handbook of Theoret. Comp. Sci., Elsevier, Amsterdam (1990), pp. 193–242.Google Scholar
  70. 70.
    R. Borie, R. Parker, and C. Tovey, “Deterministic decomposition of recursive graph classes,” SIAM J. Discrete Math., 4, 481–501 (1991).MATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    S. Arnborg, J. Lagergren, and D. Seese, “Easy problems for tree-decomposable graphs,” J. Algorithms, 12, No. 2, 308–340 (1991).MATHCrossRefMathSciNetGoogle Scholar
  72. 72.
    B. Courcelle and M. Mosbah, “Monadic second order evaluations on tree-decomposable graphs,” Theor. Comp. Sci., 109, 49–82 (1993).MATHCrossRefMathSciNetGoogle Scholar
  73. 73.
    H. L. Bodlaender, “Treewidth: algorithmic techniques and results,” in: I. Privara and P. Ruzicka (eds.), Proc. 22nd Intern. Symp. on Mathematical Foundations of Computer Science, MFCS’97, LNCS, 1295, Springer-Verlag, Berlin (1997), pp. 29–36.Google Scholar
  74. 74.
    H. M. Markowitz, “The elimination form of the inverse and its application to linear programming,” Manag. Sci., 3, 255–269 (1957).MATHMathSciNetGoogle Scholar
  75. 75.
    H. L. Bodlaender, “A linear time algorithm for finding tree-decompositions of small treewidth,” SIAM J. Computing, 25, 1305–1317 (1996).MATHCrossRefMathSciNetGoogle Scholar
  76. 76.
    H. Röhrig, “Tree decomposition: A feasibility study,” Master’s Thesis, Max-Planck-Institut für Informatik, Saarbrücken (1998).Google Scholar
  77. 77.
    D. Lapoire, “Treewidth and duality for planar hypergraphs,” Prepr. (2002) (http://citeseer.ifi.unizh.ch/472064.html).
  78. 78.
    V. Bouchitté, F. Mazoit, and I. Todinca, “Chordal embeddings of planar graphs,” Discrete Math., 273, 85–102 (2003).MATHCrossRefMathSciNetGoogle Scholar
  79. 79.
    V. Bouchitté, D. Kratsch, H. Mueller, and I. Todinca, “On treewidth approximations,” Discrete Appl. Math., 136, 183–196 (2004).MATHCrossRefMathSciNetGoogle Scholar
  80. 80.
    E. Amir, “Efficient approximations for triangulation of minimum treewidth,” Proc. 17th Conf. on Uncertainty in Artificial Intelligence (2001), pp. 7–15.Google Scholar
  81. 81.
    G. J. Woeginger, “Exact algorithms for NP-hard problems: A survey,” Combinatorial Optimization: “Eureka, You Shrink!” LNCS, 2570, Springer, Berlin (2003), pp. 185–207.Google Scholar
  82. 82.
    F. V. Fomin, D. Kratsch, and I. Todinca, “Exact (exponential) algorithms for tree-width and minimum fill-in,” in: Proc. 31st Intern. Colloquium on Automata, Languages and Programming, LNCS, 3124, Springer Verlag, Berlin (2004), pp. 568–580.Google Scholar
  83. 83.
    S. Arnborg and A. Proskurowski, “Characterization and recognition of partial 3-trees,” SIAM J. Alg. Discrete Math., 7, 305–314 (1986).MATHMathSciNetGoogle Scholar
  84. 84.
    H. L. Bodlaender, A. M. C. A. Koster, F. van den Eijkhof, and L. C. van der Gaag, “Preprocessing for triangulation of probabilistic networks,” in: J. Breese and D. Koller (eds.), Proc. 17th Conf. on Uncertainty in Artificial Intelligence, Morgan Kaufmann, San Francisco (2001), pp. 32–39.Google Scholar
  85. 85.
    H. L. Bodlaender and A. M. C. A. Koster, “Safe separators for treewidth,” Proc. 6th Workshop on Algorithm Engineering and Experiments ALENEX04 (2004), pp. 70–78.Google Scholar
  86. 86.
    K. Shoikhet and D. Geiger, “A practical algorithm for finding optimal triangulations,” in: Proc. Nat. Conf. on Artificial Intelligence AAAI’97, Morgan Kaufmann, San Mateo (1997), pp. 185–190.Google Scholar
  87. 87.
    V. Gogate and R. Dechter, “A complete anytime algorithm for treewidth,” Proc. 20th Conf. on Uncertainty in Artificial Intelligence (2004) (http://citeseer.ist.psu.edu/gogate04complete.html).
  88. 88.
    P. Heggernes, J. A. Telle, and Y. Villanger, “Computing minimal triangulations in time O(n α log n)=o(n 2.376),” SIAM J. Discrete Math., 19, No. 4, 900–913 (2005).MATHCrossRefMathSciNetGoogle Scholar
  89. 89.
    J. R. S. Blair, P. Heggernes, and J. A. Telle, “A practical algorithm for making filled graphs minimal,” Theor. Comp. Sci., 250, 125–141 (2001).MATHCrossRefMathSciNetGoogle Scholar
  90. 90.
    E. Dahlhaus, “Minimal elimination ordering inside a given chordal graph,” Proc. 3nd Intern. Workshop on Graph-Theoretic Concepts in Computer Science WG’97, LNCS, 1335, Springer Verlag, Berlin-Heidelberg (1997), pp. 132–143.CrossRefGoogle Scholar
  91. 91.
    P. Heggernes and Y. Villanger, “Efficient implementation of a minimal triangulation algorithm,” in: R. Möhring and R. Raman (eds.), Proc. 10th Ann. Eur. Symp. on Algorithms, ESA’2002, LNCS, 2461, Springer Verlag, Berlin-Heidelberg (2002), pp. 550–561.Google Scholar
  92. 92.
    I. V. Hicks, A. M. C. A. Koster, and E. Kolotoglu, “Branch and tree decomposition techniques for discrete optimization,” Tutorials Oper. Res., INFORMS, New Orleans (2005) (http://ie.tamu.edu/People/faculty/ Hicks/bwtw.pdf).
  93. 93.
    E. Bachoore and H. L. Bodlaender, “New upper bound heuristics for tree-width,” in: S. E. Nikoletseas (ed.), Proc. 4th Intern. Workshop on Experimental and Efficient Algorithms WEA 2005, LNCS, 3503, Springer-Verlag, Berlin (2005), pp. 217–227.Google Scholar
  94. 94.
    F. Clautiaux, J. Carlier, A. Moukrim, and S. Negre, “New lower and upper bounds for graph treewidth,” in: J. D. P. Rolim (ed.), Proc. Intern. Workshop on Experimental and Efficient Algorithms, WEA 2003, LNCS, 2647, Springer-Verlag, Berlin (2003), pp. 70–80.Google Scholar
  95. 95.
    F. Clautiaux, A. Moukrim, S. Négre, and J. Carlier, “Heuristic and meta-heuristic methods for computing graph treewidth,” RAIRO Oper. Res., 38, 13–26 (2004).MATHCrossRefMathSciNetGoogle Scholar
  96. 96.
    A. M. C. A. Koster, “Frequency assignment — models and algorithms,” PhD Thesis, Maastricht Univ., Maastricht (1999).Google Scholar
  97. 97.
    U. Kjaerulff, “Optimal decomposition of probabilistic networks by simulated annealing,” Statistics and Computing, 2, 2–17 (1992).CrossRefGoogle Scholar
  98. 98.
    P. Larranaga, C. M. H. Kuijpers, M. Roza, and R. H. Murga, “Decomposing Bayesian networks: Triangulation of the moral graph with genetic algorithms,” Statistics and Computing, 7, No. 1, 19–34 (1997).CrossRefGoogle Scholar
  99. 99.
    B. Lucena, “A new lower bound for tree-width using maximum cardinality search,” SIAM J. Discrete Math., 16, 345–353 (2003).MATHCrossRefMathSciNetGoogle Scholar
  100. 100.
    H. L. Bodlaender, A. M. C. A. Koster, and T. Wolle, “Contraction and treewidth lower bounds,” in: S. Albers and T. Radzik (eds.), Proc. 12th Ann. Eur. Symp. on Algorithms, ESA2004, LNCS, 3221, Springer, Berlin (2004), pp. 628–639.Google Scholar
  101. 101.
    T. Wolle, A. M. C. A. Koster, and H. L. Bodlaender, “A note on contraction degeneracy,” Techn. Report UU-CS-2004-042, Utrecht Univ., Inst. of Inform. and Comput. Sci., Utrecht (2004).Google Scholar

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© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • O. A. Shcherbina
    • 1
  1. 1.Faculty of MathematicsUniversity of ViennaAustria

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