Heuristic Computation of Exact Treewidth

Author Hisao Tamaki

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Hisao Tamaki
  • Department of Computer Science, Meiji University, Tokyo, Japan


I thank Holger Dell for posing the challenging bonus instances, which have kept defying my "great ideas", showing how they fail, and pointing to yet greater ideas.

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Hisao Tamaki. Heuristic Computation of Exact Treewidth. In 20th International Symposium on Experimental Algorithms (SEA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 233, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We are interested in computing the treewidth tw(G) of a given graph G. Our approach is to design heuristic algorithms for computing a sequence of improving upper bounds and a sequence of improving lower bounds, which would hopefully converge to tw(G) from both sides. The upper bound algorithm extends and simplifies the present author’s unpublished work on a heuristic use of the dynamic programming algorithm for deciding treewidth due to Bouchitté and Todinca. The lower bound algorithm is based on the well-known fact that, for every minor H of G, we have tw(H) ≤ tw(G). Starting from a greedily computed minor H_0 of G, the algorithm tries to construct a sequence of minors H_0, H_1, ..., H_k with tw(H_i) < tw(H_{i + 1}) for 0 ≤ i < k and hopefully tw(H_k) = tw(G). We have implemented a treewidth solver based on this approach and have evaluated it on the bonus instances from the exact treewidth track of PACE 2017 algorithm implementation challenge. The results show that our approach is extremely effective in tackling instances that are hard for conventional solvers. Our solver has an additional advantage over conventional ones in that it attaches a compact certificate to the lower bound it computes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • graph algorithm
  • treewidth
  • heuristics
  • BT dynamic programming
  • contraction
  • obstruction
  • minimal forbidden minor
  • certifying algorithms


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  1. Ernst Althaus, Daniela Schnurbusch, Julian Wüschner, and Sarah Ziegler. On tamaki’s algorithm to compute treewidths. In 19th International Symposium on Experimental Algorithms (SEA 2021). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2021. Google Scholar
  2. Stefan Arnborg, Derek G Corneil, and Andrzej Proskurowski. Complexity of finding embeddings in a k-tree. SIAM Journal on Algebraic Discrete Methods, 8(2):277-284, 1987. Google Scholar
  3. Max Bannach, Sebastian Berndt, and Thorsten Ehlers. Jdrasil: A modular library for computing tree decompositions. In 16th International Symposium on Experimental Algorithms (SEA 2017). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  4. Anne Berry, Pinar Heggernes, and Genevieve Simonet. The minimum degree heuristic and the minimal triangulation process. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 58-70. Springer, 2003. Google Scholar
  5. Hans L Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on computing, 25(6):1305-1317, 1996. Google Scholar
  6. Hans L Bodlaender. Treewidth: characterizations, applications, and computations. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 1-14. Springer, 2006. Google Scholar
  7. Hans L Bodlaender and Arie MCA Koster. Safe separators for treewidth. Discrete Mathematics, 306(3):337-350, 2006. Google Scholar
  8. Hans L Bodlaender and Arie MCA Koster. Treewidth computations i. upper bounds. Information and Computation, 208(3):259-275, 2010. Google Scholar
  9. Hans L Bodlaender and Arie MCA Koster. Treewidth computations ii. lower bounds. Information and Computation, 209(7):1103-1119, 2011. Google Scholar
  10. Vincent Bouchitté and Ioan Todinca. Treewidth and minimum fill-in: Grouping the minimal separators. SIAM Journal on Computing, 31(1):212-232, 2001. Google Scholar
  11. Holgar Dell. PACE-challenge/Treewidth. https://github.com/PACE-challenge/Treewidth, 2017. [github repository, accessed January 12, 2022].
  12. Holgar Dell. Treewidth-PACE-2017-bonus-instances. https://github.com/PACE-challenge/Treewidth-PACE-2017-bonus-instances/, 2017. [github repository, accessed January 12, 2022].
  13. Holger Dell, Christian Komusiewicz, Nimrod Talmon, and Mathias Weller. The pace 2017 parameterized algorithms and computational experiments challenge: The second iteration. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  14. Pinar Heggernes. Minimal triangulations of graphs: A survey. Discrete Mathematics, 306(3):297-317, 2006. Google Scholar
  15. Neil Robertson and Paul D. Seymour. Graph minors. ii. algorithmic aspects of tree-width. Journal of algorithms, 7(3):309-322, 1986. Google Scholar
  16. Neil Robertson and Paul D Seymour. Graph minors. xiii. the disjoint paths problem. Journal of combinatorial theory, Series B, 63(1):65-110, 1995. Google Scholar
  17. Neil Robertson and Paul D Seymour. Graph minors. xx. wagner’s conjecture. Journal of Combinatorial Theory, Series B, 92(2):325-357, 2004. Google Scholar
  18. Hisao Tamaki. Computing treewidth via exact and heuristic lists of minimal separators. In International Symposium on Experimental Algorithms, pages 219-236. Springer, 2019. Google Scholar
  19. Hisao Tamaki. A heuristic use of dynamic programming to upperbound treewidth. arXiv preprint, 2019. URL: http://arxiv.org/abs/1909.07647.
  20. Hisao Tamaki. Positive-instance driven dynamic programming for treewidth. Journal of Combinatorial Optimization, 37(4):1283-1311, 2019. Google Scholar
  21. Hisao Tamaki. A heuristic for listing almost-clique minimal separators of a graph. arXiv preprint, 2021. URL: http://arxiv.org/abs/2108.07551.
  22. Hisao Tamaki. twalgor/tw. https://github.com/twalgor/, 2022. [github repository].
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