Document Open Access Logo

Polynomial Delay Algorithm for Minimal Chordal Completions

Authors Caroline Brosse, Vincent Limouzy, Arnaud Mary

Thumbnail PDF


  • Filesize: 0.74 MB
  • 16 pages

Document Identifiers

Author Details

Caroline Brosse
  • Université Clermont Auvergne, Clermont Auvergne INP, CNRS, Mines Saint-Etienne, Limos, F-63000 Clermont-Ferrand, France
Vincent Limouzy
  • Université Clermont Auvergne, Clermont Auvergne INP, CNRS, Mines Saint-Etienne, Limos, F-63000 Clermont-Ferrand, France
Arnaud Mary
  • Université de Lyon, Université Lyon 1, CNRS, Laboratoire de Biométrie et Biologie Evolutive UMR 5558, 69622 Villeurbanne, France
  • ERABLE team, Inria Grenoble Rhône‑Alpes, Villeurbanne, France

Cite AsGet BibTex

Caroline Brosse, Vincent Limouzy, and Arnaud Mary. Polynomial Delay Algorithm for Minimal Chordal Completions. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 33:1-33:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


Motivated by the problem of enumerating all tree decompositions of a graph, we consider in this article the problem of listing all the minimal chordal completions of a graph. In [Carmeli et al., 2020] (Pods 2017) Carmeli et al. proved that all minimal chordal completions or equivalently all proper tree decompositions of a graph can be listed in incremental polynomial time using exponential space. The total running time of their algorithm is quadratic in the number of solutions and the existence of an algorithm whose complexity depends only linearly on the number of solutions remained open. We close this question by providing a polynomial delay algorithm to solve this problem which, moreover, uses polynomial space. Our algorithm relies on Proximity Search, a framework recently introduced by Conte and Uno [Conte and Uno, 2019] (Stoc 2019) which has been shown powerful to obtain polynomial delay algorithms, but generally requires exponential space. In order to obtain a polynomial space algorithm for our problem, we introduce a new general method called canonical path reconstruction to design polynomial delay and polynomial space algorithms based on proximity search.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Graph algorithms
  • Graph Algorithm
  • Algorithmic Enumeration
  • Minimal chordal completions


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Stefand Arnbog, Derek G. Corneil, and Andrzej Proskurowski. Complexity of finding embeddings in a k-tree. SIAM Journal on Algebraic Discrete Methods, 8:277-284, 1987. URL:
  2. David Avis and Komei Fukuda. Reverse search for enumeration. Discrete applied mathematics, 65(1-3):21-46, 1996. Google Scholar
  3. Anne Berry, Jean RS Blair, Pinar Heggernes, and Barry W Peyton. Maximum cardinality search for computing minimal triangulations of graphs. Algorithmica, 39(4):287-298, 2004. Google Scholar
  4. Endre Boros, Benny Kimelfeld, Reinhard Pichler, and Nicole Schweikardt. Enumeration in data management (dagstuhl seminar 19211). Technical report, Dagstuhl Seminar, 2019. URL:
  5. Vincent Bouchitté and Ioan Todinca. Minimal triangulations for graphs with “few” minimal separators. In European Symposium on Algorithms, pages 344-355. Springer, 1998. Google Scholar
  6. Caroline Brosse, Aurélie Lagoutte, Vincent Limouzy, Arnaud Mary, and Lucas Pastor. Efficient enumeration of maximal split subgraphs and sub-cographs and related classes. arXiv preprint, 2020. URL:
  7. Nofar Carmeli, Batya Kenig, and Benny Kimelfeld. Efficiently enumerating minimal triangulations. In Emanuel Sallinger, Jan Van den Bussche, and Floris Geerts, editors, Proceedings of the 36th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2017, Chicago, IL, USA, May 14-19, 2017, pages 273-287. ACM, 2017. URL:
  8. Nofar Carmeli, Batya Kenig, Benny Kimelfeld, and Markus Kröll. Efficiently enumerating minimal triangulations. Discrete Applied Mathematics, In Press, 2020. Google Scholar
  9. Sara Cohen, Benny Kimelfeld, and Yehoshua Sagiv. Generating all maximal induced subgraphs for hereditary and connected-hereditary graph properties. Journal of Computer and System Sciences, 74(7):1147-1159, November 2008. URL:
  10. Alessio Conte, Roberto Grossi, Andrea Marino, and Luca Versari. Listing maximal subgraphs satisfying strongly accessible properties. SIAM Journal on Discrete Mathematics, 33(2):587-613, 2019. Google Scholar
  11. Alessio Conte, Andrea Marino, Roberto Grossi, Takeaki Uno, and Luca Versari. Proximity Search For Maximal Subgraph Enumeration. arXiv:1912.13446 [cs], July 2020. URL:
  12. Alessio Conte and Takeaki Uno. New polynomial delay bounds for maximal subgraph enumeration by proximity search. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 1179-1190, 2019. Google Scholar
  13. G. A. Dirac. On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 25:71-76, 1961. URL:
  14. Rudolf Halin. S-functions for graphs. Journal of Geometry, 8:171-186, 1976. URL:
  15. Pinar Heggernes. Minimal triangulations of graphs: a survey. Discrete Mathematics, 306(3):297-317, 2006. Google Scholar
  16. Pinar Heggernes and Charis Papadopoulos. Single-edge monotonic sequences of graphs and linear-time algorithms for minimal completions and deletions. Theoretical Computer Science, 410(1):1-15, 2009. Google Scholar
  17. David S Johnson, Mihalis Yannakakis, and Christos H Papadimitriou. On generating all maximal independent sets. Information Processing Letters, 27(3):119-123, 1988. Google Scholar
  18. E. Lawler, J. Lenstra, and A. Kan. Generating all maximal independent sets: Np-hardness and polynomial-time algorithms. SIAM J. Comput., 9:558-565, 1980. Google Scholar
  19. Tatsuo Ohtsuki. A fast algorithm for finding an optimal ordering for vertex elimination on a graph. SIAM Journal on Computing, 5(1):133-145, 1976. Google Scholar
  20. Andreas Parra and Petra Scheffler. Characterizations and algorithmic applications of chordal graph embeddings. Discrete Applied Mathematics, 79(1-3):171-188, 1997. Google Scholar
  21. Noam Ravid, Dori Medini, and Benny Kimelfeld. Ranked enumeration of minimal triangulations. In Proceedings of the 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS '19, pages 74-88, New York, NY, USA, 2019. Association for Computing Machinery. URL:
  22. Neil Robertson and P.D Seymour. Graph minors. iii. planar tree-width. Journal of Combinatorial Theory, Series B, 36(1):49-64, 1984. URL:
  23. Donald J Rose, R Endre Tarjan, and George S Lueker. Algorithmic aspects of vertex elimination on graphs. SIAM Journal on computing, 5(2):266-283, 1976. Google Scholar
  24. Ioan Todinca. Aspects algorithmiques des triangulations minimales des graphes. PhD thesis, École normale supérieure (Lyon), 1999. Google Scholar
  25. Shuji Tsukiyama, Mikio Ide, Hiromu Ariyoshi, and Isao Shirakawa. A New Algorithm for Generating All the Maximal Independent Sets. SIAM J. Comput., 6(3):505-517, September 1977. URL:
  26. Takeaki Uno. Two general methods to reduce delay and change of enumeration algorithms. National Intsitute of Informatics, Tech. report, E 4, 2003. URL:
  27. Mihalis Yannakakis. Computing the minimum fill-in is np-complete. SIAM Journal on Algebraic and Discrete Methods, 2:77-79, 1981. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail