Abstract
While the theoretical aspects concerning the computation of tree width – one of the most important graph parameters – are well understood, it is not clear how it can be computed practically. As tree width has a wide range of applications, e. g. in bioinformatics or artificial intelligence, this lack of understanding hinders the applicability of many important algorithms in the real world. The Parameterized Algorithms and Computational Experiments (PACE) challenge therefore chose the computation of tree width as one of its challenge problems in 2016 and again in 2017. In 2016, Hisao Tamaki (Meiji University) presented a new algorithm that outperformed the other approaches (including SAT solvers and branch-and-bound) by far. An implementation of Tamaki’s algorithm allowed Larisch (King-Abdullah University of Science and Engineering) and Salfelder (University of Leeds) to solve over 50% of the test suite of PACE 2017 (containing graphs with over 3500 nodes) in under six seconds (and the remaining 50% in under 30 min). Before PACE 2016, no algorithm was known to compute tree width on graphs with about 100 nodes. As a wide range of parameterized algorithms require the computation of a tree decomposition as a first step, this breakthrough result allows practical implementations of these algorithms for the first time.
This work starts with a gentle introduction to tree width and its use in parameterized complexity, followed by an algorithmic approach for the exact computation of the tree width of a graph, based on a variant of the well-studied cops-and-robber game. Finally, we present a streamlined version of Tamaki’s algorithms due to Bannach and Berndt based on this game.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
All graphs in this work are undirected, unless stated otherwise.
References
Bannach, M., Berndt, S.: Positive-Instance Driven Dynamic Programming for Graph Searching, unpublished
Bannach, M., Berndt, S., Ehlers, T.: Jdrasil: a modular library for computing tree decompositions. In: Proceedings of SEA. LIPIcs, vol. 75, pp. 28:1–28:21. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017)
Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. In: Proceedings of STOC, pp. 226–234. ACM (1993)
Bodlaender, H.L., Koster, A.M.C.A.: Treewidth computations I. Upper bounds. Inf. Comput. 208(3), 259–275 (2010)
Bouchitté, V., Todinca, I.: Listing all potential maximal cliques of a graph. Theor. Comput. Sci. 276(1–2), 17–32 (2002)
Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-319-21275-3
Dell, H., Husfeldt, T., Jansen, B.M.P., Kaski, P., Komusiewicz, C., Rosamond, F.A.: The first parameterized algorithms and computational experiments challenge. In: Proceedings of IPEC. LIPIcs, vol. 63, pp. 30:1–30:9. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)
Dell, H., Komusiewicz, C., Talmon, N., Weller, M.: The PACE 2017 parameterized algorithms and computational experiments challenge: the second iteration. In: Proceedings of IPEC. LIPIcs, pp. 30:1–30:12. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018)
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, London (2016). https://doi.org/10.1007/978-1-4471-5559-1
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (2012). https://doi.org/10.1007/978-1-4612-0515-9
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-29953-X
Karger, D.R., Srebro, N.: Learning markov networks: maximum bounded tree-width graphs. In: Proceedings of SODA, pp. 392–401. ACM/SIAM (2001)
Larisch, L., Salfelder, F.: p17 (2017). https://github.com/freetdi/p17. Accessed 1 Mar 2018
Seymour, P.D., Thomas, R.: Graph searching and a min-max theorem for tree-width. J. Comb. Theory Ser. B 58, 22–33 (1993)
Song, Y., Liu, C., Malmberg, R.L., Pan, F., Cai, L.: Tree decomposition based fast search of RNA structures including pseudoknots in genomes. In: Proceedings of CSB, pp. 223–234. IEEE Computer Society (2005)
Tamaki, H.: Treewidth-exact (2016). https://github.com/TCS-Meiji/treewidth-exact. Accessed 1 Mar 2018
Tamaki, H.: Positive-instance driven dynamic programming for treewidth. In: Proceedings of ESA. LIPIcs, vol. 87, pp. 68:1–68:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017)
Tamaki, H., Ohtsuka, H.: tw-exact (2017). https://github.com/TCS-Meiji/PACE2017-TrackA. Accessed 1 Mar 2018
Acknowledgments
The author likes to thank Max Bannach for many fruitful discussions around theoretical and practical approaches to tree width.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Berndt, S. (2018). Computing Tree Width: From Theory to Practice and Back. In: Manea, F., Miller, R., Nowotka, D. (eds) Sailing Routes in the World of Computation. CiE 2018. Lecture Notes in Computer Science(), vol 10936. Springer, Cham. https://doi.org/10.1007/978-3-319-94418-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-94418-0_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94417-3
Online ISBN: 978-3-319-94418-0
eBook Packages: Computer ScienceComputer Science (R0)