Better algorithms for the pathwidth and treewidth of graphs

  • Hans L. Bodlaender
  • Ton Kloks
Algorithms (Session 13)
Part of the Lecture Notes in Computer Science book series (LNCS, volume 510)


In this paper we give, for all constants k, explicit O(n log2n) algorithms, that given a graph G = (V,E), decide whether the treewidth (or pathwidth) of G is at most k, and if so, find a tree-decomposition or (path-decomposition) of G with treewidth (or pathwidth) at most k. In contrast with previous solutions, our algorithms do not rely on non-constructive reasoning, and are single exponential in k. This result implies a similar result for several graph notions that are equivalent with treewidth or pathwidth.


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  1. [1]
    S. Arnborg. Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey. BIT, 25:2–23, 1985.Google Scholar
  2. [2]
    S. Arnborg, D. G. Corneil, and A. Proskurowski. Complexity of finding embeddings in a k-tree. SIAM J. Alg. Disc. Meth., 8:277–284, 1987.Google Scholar
  3. [3]
    S. Arnborg, B. Courcelle, A. Proskurowski, and D. Seese. An algebraic theory of graph reduction. Technical Report 90-02, Laboratoire Bordelais de Recherche en Informatique, Bordeaux, 1990. To appear in Proceedings 4th Workshop on Graph Grammars and Their Applications to Computer Science.Google Scholar
  4. [4]
    S. Arnborg, J. Lagergren, and D. Seese. Problems easy for tree-decomposable graphs (extended abstract). In Proceedings of the 15'th International Colloquium on Automata, Languages and Programming, pages 38–51. Springer Verlag, Lect. Notes in Comp. Sc. 317, 1988. To appear in J. of Algorithms.Google Scholar
  5. [5]
    S. Arnborg and A. Proskurowski. Characterization and recognition of partial 3-trees. SIAM J. Alg. Disc. Meth., 7:305–314, 1986.Google Scholar
  6. [6]
    S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Disc. Appl. Math., 23:11–24, 1989.Google Scholar
  7. [7]
    M. W. Bern, E. L. Lawler, and A. L. Wong. Linear time computation of optimal subgraphs of decomposable graphs. J. Algorithms, 8:216–235, 1987.Google Scholar
  8. [8]
    H. L. Bodlaender. Classes of graphs with bounded treewidth. Technical Report RUU-CS-86-22, Dept. of Computer Science, Utrecht University, Utrecht, 1986.Google Scholar
  9. [9]
    H. L. Bodlaender. Dynamic programming algorithms on graphs with bounded tree-width. In Proceedings of the 15'th International Colloquium on Automata, Languages and Programming, pages 105–119. Springer Verlag, Lecture Notes in Computer Science, vol. 317, 1988.Google Scholar
  10. [10]
    H. L. Bodlaender. NC-algorithms for graphs with small treewidth. In J. van Leeuwen, editor, Proc. Workshop on Graph-Theoretic Concepts in Computer Science WG'88, pages 1–10. Springer Verlag, LNCS 344, 1988.Google Scholar
  11. [11]
    H. L. Bodlaender. Complexity of path forming games. Technical Report RUU-CS-89-29, Utrecht University, Utrecht, 1989.Google Scholar
  12. [12]
    H. L. Bodlaender. Improved self-reduction algorithms for graphs with bounded treewidth. In Proc. 15th Int. Workshop on Graph-theoretic Concepts in Computer Science WG'89, pages 232–244. Springer Verlag, Lect. Notes in Computer Science, vol. 411, 1990. To appear in: Annals of Discrete Mathematics.Google Scholar
  13. [13]
    R. B. Borie, R. G. Parker, and C. A. Tovey. Automatic generation of linear algorithms from predicate calculus descriptions of problems on recursive constructed graph families. Manuscript, 1988.Google Scholar
  14. [14]
    N. Chandrasekharan and S. T. Hedetniemi. Fast parallel algorithms for tree decomposing and parsing partial k-trees. In Proc. 26th Annual Allerton Conference on Communication, Control, and Computing, Urbana-Champaign, Illinois, 1988.Google Scholar
  15. [15]
    B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation, 85:12–75, 1990.Google Scholar
  16. [16]
    I. S. Duff and J. K. Reid. The multifrontal solution of indefinite sparse symmetric linear equations. ACM Transactions on Mathematical Software, 9:302–325, 1983.Google Scholar
  17. [17]
    J. A. Ellis, I. H. Sudborough, and J. Turner. Graph separation and search number. Report DCS-66-IR, University of Victoria, 1987.Google Scholar
  18. [18]
    M. R. Fellows and K. R. Abrahamson. Cutset regularity beats well-quasi-ordering for bounded treewidth. Manuscript, 1990.Google Scholar
  19. [19]
    M. R. Fellows and M. A. Langston. An analogue of the Myhill-Nerode theorem and its use in computing finite-basis characterizations. In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, pages 520–525, 1989.Google Scholar
  20. [20]
    M. R. Fellows and M. A. Langston. On search, decision and the efficiency of polynomial-time algorithms. In Proceedings of the 21th Annual Symposium on Theory of Computing, pages 501–512, 1989.Google Scholar
  21. [21]
    A. Habel. Hyperedge Replacement: Grammars and Languages. PhD thesis, Univ. Bremen, 1988.Google Scholar
  22. [22]
    J. Lagergren. Efficient parallel algorithms for tree-decomposition and related problems. In Proceedings of the 31th Annual Symposium on Foundations of Computer Science, pages 173–182, 1990.Google Scholar
  23. [23]
    J. Lagergren. Algorithms and Minimal Forbidden Minors for Tree-decomposable Graphs. PhD thesis, Royal Institute of Technology, Stockholm, Sweden, 1991.Google Scholar
  24. [24]
    J. Lagergren and S. Arnborg. Finding minimal forbidden minors using a finite congruence. To appear in: proceedings ICALP'91.Google Scholar
  25. [25]
    S. Lauritzen and D. Spiegelhalter. 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.Google Scholar
  26. [26]
    J. Matousěk and R. Thomas. Algorithms finding tree-decompositions of graphs. J. Algorithms, 12:1–22, 1991.Google Scholar
  27. [27]
    R. H. Möhring. Graph problems related to gate matrix layout and PLA folding. Technical Report 223/1989, Technical University of Berlin, 1989.Google Scholar
  28. [28]
    N. Robertson and P. D. Seymour. Graph minors — a survey. In I. Anderson, editor, Surveys in Combinatorics, pages 153–171. Cambridge Univ. Press, 1985.Google Scholar
  29. [29]
    P. Scheffler. Linear-time algorithms for NP-complete problems restricted to partial k-trees. Report R-MATH-03/87, Karl-Weierstrass-Institut Für Mathematik, Berlin, GDR, 1987.Google Scholar
  30. [30]
    L. C. van der Gaag. Probability-Based Models for Plausible Reasoning. PhD thesis, University of Amsterdam, 1990.Google Scholar
  31. [31]
    T. V. Wimer. Linear algorithms on k-terminal graphs. PhD thesis, Dept. of Computer Science, Clemson University, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Hans L. Bodlaender
    • 1
  • Ton Kloks
    • 1
  1. 1.Department of Computer ScienceUtrecht UniversityUtrechtthe Netherlands

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