Skip to main content
SpringerLink
Log in
Book cover

Treewidth, Kernels, and Algorithms pp 189–213Cite as

Computing Tree Decompositions

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12160))

Abstract

In this chapter we review the most important algorithmic approaches to the following problem: given a graph G, compute a tree decomposition of G of (nearly) optimum width. We present the 4-approximation algorithm running in time \(\mathcal {O}(27^k\cdot k^2\cdot n^2)\), which was first proposed by Robertson and Seymour in the Graph Minors series, and we discuss the main ideas behind the exact algorithm of Bodlaender that runs in linear fixed-parameter time [2].

The work of Mi. Pilipczuk on this article is supported by the project TOTAL that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 677651).

figure a

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    We note that Bodlaender in [2] uses a slightly different notion of the improved graph, where we require that the number of common neighbors of u and v is more than k; this essentially restricts attention to looking at paths of length 2.

References

  1. Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a \(k\)-tree. SIAM J. Discrete Math. 8(2), 277–284 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: A \(c^k n\)\(5\)-approximation algorithm for treewidth. SIAM J. Comput. 45(2), 317–378 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bodlaender, H.L., Kloks, T.: Better algorithms for the pathwidth and treewidth of graphs. In: Albert, J.L., Monien, B., Artalejo, M.R. (eds.) ICALP 1991. LNCS, vol. 510, pp. 544–555. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-54233-7_162

    Chapter  MATH  Google Scholar 

  5. Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21(2), 358–402 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bodlaender, H.L., Thilikos, D.M.: Constructive linear time algorithms for branchwidth. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 627–637. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-63165-8_217

    Chapter  Google Scholar 

  7. Bojańczyk, M., Pilipczuk, M.: Definability equals recognizability for graphs of bounded treewidth. CoRR, abs/1605.03045 (2016). Extended abstract published in the proceedings of LICS 2016

    Google Scholar 

  8. Bojańczyk, M., Pilipczuk, M.: Optimizing tree decompositions in MSO. In: Proceedings of the 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017. LIPIcs, vol. 66, pp. 15:1–15:13. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

    Google Scholar 

  9. Courcelle, B., Lagergren, J.: Equivalent definitions of recognizability for sets of graphs of bounded tree-width. Math. Struct. Comput. Sci. 6(2), 141–165 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3

    Book  MATH  Google Scholar 

  11. Dell, H., Komusiewicz, C., Talmon, N., Weller, M.: The PACE 2017 parameterized algorithms and computational experiments challenge: the second iteration. In: 12th International Symposium on Parameterized and Exact Computation, IPEC 2017, LIPIcs, vol. 89, pp. 30:1–30:12. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

    Google Scholar 

  12. Elberfeld, M., Schweitzer, P.: Canonizing graphs of bounded tree width in logspace. ACM Trans. Comput. Theory (TOCT) 9(3), 1–29 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Feige, U., Hajiaghayi, M., Lee, J.R.: Improved approximation algorithms for minimum weight vertex separators. SIAM J. Comput. 38(2), 629–657 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Flum, J., Frick, M., Grohe, M.: Query evaluation via tree-decompositions. J. ACM 49(6), 716–752 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)

    Google Scholar 

  16. Giannopoulou, A.C., Pilipczuk, M., Raymond, J., Thilikos, D.M., Wrochna, M.: Cutwidth: obstructions and algorithmic aspects. Algorithmica 81(2), 557–588 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jeong, J., Kim, E.J., Oum, S.I.: Constructive algorithm for path-width of matroids. In: Krauthgamer, R. (ed.) Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, pp. 1695–1704. SIAM (2016)

    Google Scholar 

  18. Jeong, J., Kim, E.J., Oum, S.: Finding branch-decompositions of matroids, hypergraphs, and more. In: Proceedings of the 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018. LIPIcs, vol. 107, pp. 80:1–80:14. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

    Google Scholar 

  19. Kleinberg, J.M., Tardos, É.: Algorithm Design. Addison-Wesley (2006)

    Google Scholar 

  20. Lagergren, J.: Upper bounds on the size of obstructions and intertwines. J. Comb. Theory Ser. B 73(1), 7–40 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lagergren, J., Arnborg, S.: Finding minimal forbidden minors using a finite congruence. In: Albert, J.L., Monien, B., Artalejo, M.R. (eds.) ICALP 1991. LNCS, vol. 510, pp. 532–543. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-54233-7_161

    Chapter  Google Scholar 

  22. Lokshtanov, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth. SIAM J. Comput. 46(1), 161–189 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Oum, S., Seymour, P.D.: Approximating clique-width and branch-width. J. Comb. Theory Ser. B 96(4), 514–528 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Raghavendra, P., Steurer, D.: Graph expansion and the unique games conjecture. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, pp. 755–764. ACM (2010)

    Google Scholar 

  25. Reed, B.A.: Finding approximate separators and computing tree width quickly. In: Proceedings of the 24th Annual ACM Symposium on Theory of Computing, STOC 1992, pp. 221–228. ACM (1992)

    Google Scholar 

  26. Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory Ser. B 63(1), 65–110 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  27. Robertson, N., Seymour, P.D.: Graph minors. XX. Wagner’s conjecture. J. Comb. Theory Ser. B 92(2), 325–357 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Thilikos, D.M., Serna, M.J., Bodlaender, H.L.: Constructive linear time algorithms for small cutwidth and carving-width. In: Goos, G., Hartmanis, J., van Leeuwen, J., Lee, D.T., Teng, S.-H. (eds.) ISAAC 2000. LNCS, vol. 1969, pp. 192–203. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-40996-3_17

    Chapter  MATH  Google Scholar 

  29. Thilikos, D.M., Serna, M.J., Bodlaender, H.L.: Cutwidth I: a linear time fixed parameter algorithm. J. Algorithms 56(1), 1–24 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Thilikos, D.M., Serna, M.J., Bodlaender, H.L.: Cutwidth II: algorithms for partial \(w\)-trees of bounded degree. J. Algorithms 56(1), 25–49 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wu, Y., Austrin, P., Pitassi, T., Liu, D.: Inapproximability of treewidth and related problems. J. Artif. Intell. Res. 49, 569–600 (2014)

    Article  MATH  Google Scholar 

  32. Yannakakis, M.: Computing the minimum fill-in is NP-complete. SIAM J. Algebr. Discrete Methods 2(1), 77–79 (1981)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Informatics, University of Warsaw, Warsaw, Poland

    Michał Pilipczuk

Authors
  1. Michał Pilipczuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michał Pilipczuk .

Editor information

Editors and Affiliations

  1. University of Bergen, Bergen, Norway

    Fedor V. Fomin

  2. Humboldt-University, Berlin, Germany

    Stefan Kratsch

  3. Utrecht University, Utrecht, The Netherlands

    Erik Jan van Leeuwen

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Pilipczuk, M. (2020). Computing Tree Decompositions. In: Fomin, F.V., Kratsch, S., van Leeuwen, E.J. (eds) Treewidth, Kernels, and Algorithms. Lecture Notes in Computer Science(), vol 12160. Springer, Cham. https://doi.org/10.1007/978-3-030-42071-0_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-42071-0_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-42070-3

  • Online ISBN: 978-3-030-42071-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation