Acquiring Maps of Interrelated Conjectures on Sharp Bounds

Authors Nicolas Beldiceanu, Jovial Cheukam-Ngouonou, Rémi Douence, Ramiz Gindullin, Claude-Guy Quimper

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Author Details

Nicolas Beldiceanu
  • IMT Atlantique, LS2N (TASC), Nantes, France
Jovial Cheukam-Ngouonou
  • IMT Atlantique, LS2N (TASC), Nantes, France, and Université Laval, Québec, Canada
Rémi Douence
  • IMT Atlantique, LS2N, Inria, (Gallinette), Nantes, France
Ramiz Gindullin
  • IMT Atlantique, LS2N (TASC), Nantes, France
Claude-Guy Quimper
  • Université Laval, Québec, Canada


Thanks to Hervé Grall for his participation in the definition of the map concept, and to Samir Loudni and Helmut Simonis for their comments on a preliminary version of this paper.

Cite AsGet BibTex

Nicolas Beldiceanu, Jovial Cheukam-Ngouonou, Rémi Douence, Ramiz Gindullin, and Claude-Guy Quimper. Acquiring Maps of Interrelated Conjectures on Sharp Bounds. In 28th International Conference on Principles and Practice of Constraint Programming (CP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 235, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


To automate the discovery of conjectures on combinatorial objects, we introduce the concept of a map of sharp bounds on characteristics of combinatorial objects, that provides a set of interrelated sharp bounds for these combinatorial objects. We then describe a Bound Seeker, a CP-based system, that gradually acquires maps of conjectures. The system was tested for searching conjectures on bounds on characteristics of digraphs: it constructs sixteen maps involving 431 conjectures on sharp lower and upper-bounds on eight digraph characteristics.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Heuristic function construction
  • Mathematics of computing → Combinatorial optimization
  • Acquisition of conjectures
  • digraphs
  • bounds


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  1. Mustapha Aouchiche, Gilles Caporossi, Pierre Hansen, and M. Laffay. Autographix: a survey. Electron. Notes Discret. Math., 22:515-520, 2005. URL:
  2. Nicolas Beldiceanu, Mats Carlsson, and Jean-Xavier Rampon. Global Constraint Catalog, 2nd Edition (revision a). Technical Report T2012-03, Swedish Institute of Computer Science, 2012. Available at
  3. Nicolas Beldiceanu, Mats Carlsson, Jean-Xavier Rampon, and Charlotte Truchet. Graph invariants as necessary conditions for global constraints. In Peter van Beek, editor, Principles and Practice of Constraint Programming - CP 2005, 11th International Conference, CP 2005, Sitges, Spain, October 1-5, 2005, Proceedings, volume 3709 of Lecture Notes in Computer Science, pages 92-106. Springer, 2005. Google Scholar
  4. Nicolas Beldiceanu and Helmut Simonis. A Model Seeker: Extracting Global Constraint Models from Positive Examples. In Michela Milano, editor, Principles and Practice of Constraint Programming - 18th International Conference, CP 2012, Québec City, QC, Canada, October 8-12, 2012. Proceedings, volume 7514 of Lecture Notes in Computer Science, pages 141-157. Springer, 2012. URL:
  5. Yoshua Bengio, Andrea Lodi, and Antoine Prouvost. Machine learning for combinatorial optimization: A methodological tour d'horizon. Eur. J. Oper. Res., 290(2):405-421, 2021. Google Scholar
  6. Christian Bessière, Emmanuel Hebrard, George Katsirelos, Zeynep Kızıltan, Émilie Picard-Cantin, Claude-Guy Quimper, and Toby Walsh. The balance constraint family. In Barry O'Sullivan, editor, Principles and Practice of Constraint Programming - 20th International Conference, CP 2014, Lyon, France, September 8-12, 2014. Proceedings, volume 8656 of Lecture Notes in Computer Science, pages 174-189. Springer, 2014. Google Scholar
  7. Christian Bessière, Frédéric Koriche, Nadjib Lazaar, and Barry O'Sullivan. Constraint acquisition. Artif. Intell., 244:315-342, 2017. URL:
  8. V. Brankov, P. Hansen, and D. Stevanović. Automated conjectures on upper bounds for the largest laplacian eigenvalue of graphs. Linear Algebra and its Applications, 414(2):407-424, 2006. Google Scholar
  9. Jure Brence, Ljupčo Todorovski, and Sašo Džeroski. Probabilistic grammars for equation discovery. Knowledge-Based Systems, 224:107077, 2021. URL:
  10. Céline Brouard, Simon de Givry, and Thomas Schiex. Pushing data into CP models using graphical model learning and solving. In Helmut Simonis, editor, Principles and Practice of Constraint Programming - 26th International Conference, CP 2020, Louvain-la-Neuve, Belgium, September 7-11, 2020, Proceedings, volume 12333 of Lecture Notes in Computer Science, pages 811-827. Springer, 2020. URL:
  11. John William Charnley, Simon Colton, and Ian Miguel. Automatic generation of implied constraints. In Gerhard Brewka, Silvia Coradeschi, Anna Perini, and Paolo Traverso, editors, ECAI 2006, 17th European Conference on Artificial Intelligence, August 29 - September 1, 2006, Riva del Garda, Italy, Including Prestigious Applications of Intelligent Systems (PAIS 2006), Proceedings, volume 141 of Frontiers in Artificial Intelligence and Applications, pages 73-77. IOS Press, 2006. URL:
  12. Simon Colton, Andreas Meier, Volker Sorge, and Roy L. McCasland. Automatic generation of classification theorems for finite algebras. In David A. Basin and Michaël Rusinowitch, editors, Automated Reasoning - Second International Joint Conference, IJCAR 2004, Cork, Ireland, July 4-8, 2004, Proceedings, volume 3097 of Lecture Notes in Computer Science, pages 400-414. Springer, 2004. URL:
  13. Alex Davies, Petar Veličković, Lars Buesing, Sam Blackwell, Daniel Zheng, Nenad Tomašev, Richard Tanburn, Peter Battaglia, Charles Blundell, András Juhász, Marc Lackenby, Geordie Williamson, Demis Hassabis, and Pushmeet Kohli. Advancing mathematics by guiding human intuition with ai. Nature, 600(7887):70-74, 2021. URL:
  14. Siemion Fajtlowicz. On conjectures of Graffiti. Discret. Math., 72(1-3):113-118, 1988. URL:
  15. Aaron M. Ferber, Bryan Wilder, Bistra Dilkina, and Milind Tambe. Mipaal: Mixed integer program as a layer. In The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New York, NY, USA, February 7-12, 2020, pages 1504-1511. AAAI Press, 2020. URL:
  16. Minh Hoàng Hà, Claude-Guy Quimper, and Louis-Martin Rousseau. General bounding mechanism for constraint programs. In Gilles Pesant, editor, Principles and Practice of Constraint Programming - 21st International Conference, CP 2015, Cork, Ireland, August 31 - September 4, 2015, Proceedings, volume 9255 of Lecture Notes in Computer Science, pages 158-172. Springer, 2015. Google Scholar
  17. Pierre Hansen and Gilles Caporossi. Autographix: An automated system for finding conjectures in graph theory. Electron. Notes Discret. Math., 5:158-161, 2000. URL:
  18. Samuel Kolb, Sergey Paramonov, Tias Guns, and Luc De Raedt. Learning constraints in spreadsheets and tabular data. Mach. Learn., 106(9-10):1441-1468, 2017. URL:
  19. Mohit Kumar, Stefano Teso, and Luc De Raedt. Acquiring integer programs from data. In Sarit Kraus, editor, Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI 2019, Macao, China, August 10-16, 2019, pages 1130-1136., 2019. URL:
  20. Guillaume Lample and François Charton. Deep learning for symbolic mathematics. In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020., 2020. URL:
  21. Craig E. Larson and Nicolas Van Cleemput. Automated conjecturing I: Fajtlowicz’s Dalmatian heuristic revisited. Artif. Intell., 231:17-38, 2016. URL:
  22. Jimmy Ho-Man Lee, Ka Lun Leung, and Yu Wai Shum. Consistency techniques for polytime linear global cost functions in weighted constraint satisfaction. Constraints, 19(3):270-308, 2014. Google Scholar
  23. Doug Lenat. AM: An artificial intelligence approach to discovery in mathematics. PhD thesis, Stanford University, 1976. Google Scholar
  24. Thorsten Papenbrock, Jens Ehrlich, Jannik Marten, Tommy Neubert, Jan-Peer Rudolph, Martin Schönberg, Jakob Zwiener, and Felix Naumann. Functional dependency discovery: An experimental evaluation of seven algorithms. Proc. VLDB Endow., 8(10):1082-1093, 2015. URL:
  25. Sergey Paramonov, Samuel Kolb, Tias Guns, and Luc De Raedt. Tacle: Learning constraints in tabular data. In Ee-Peng Lim, Marianne Winslett, Mark Sanderson, Ada Wai-Chee Fu, Jimeng Sun, J. Shane Culpepper, Eric Lo, Joyce C. Ho, Debora Donato, Rakesh Agrawal, Yu Zheng, Carlos Castillo, Aixin Sun, Vincent S. Tseng, and Chenliang Li, editors, Proceedings of the 2017 ACM on Conference on Information and Knowledge Management, CIKM 2017, Singapore, November 06 - 10, 2017, pages 2511-2514. ACM, 2017. URL:
  26. Anselm Paulus, Michal Rolínek, Vít Musil, Brandon Amos, and Georg Martius. Comboptnet: Fit the Right NP-Hard Problem by Learning Integer Programming Constraints, 2021. URL:
  27. Émilie Picard-Cantin, Mathieu Bouchard, Claude-Guy Quimper, and Jason Sweeney. Learning the parameters of global constraints using branch-and-bound. In J. Christopher Beck, editor, Principles and Practice of Constraint Programming - 23rd International Conference, CP 2017, Melbourne, VIC, Australia, August 28 - September 1, 2017, Proceedings, volume 10416 of Lecture Notes in Computer Science, pages 512-528. Springer, 2017. URL:
  28. Steve Prestwich. Robust constraint acquisition by sequential analysis. In Giuseppe De Giacomo, Alejandro Catalá, Bistra Dilkina, Michela Milano, Senén Barro, Alberto Bugarín, and Jérôme Lang, editors, ECAI 2020 - 24th European Conference on Artificial Intelligence, 29 August-8 September 2020, Santiago de Compostela, Spain, August 29 - September 8, 2020 - Including 10th Conference on Prestigious Applications of Artificial Intelligence (PAIS 2020), volume 325 of Frontiers in Artificial Intelligence and Applications, pages 355-362. IOS Press, 2020. URL:
  29. Gal Raayoni, Shahar Gottlieb, Yahel Manor, George Pisha, Yoav Harris, Uri Mendlovic, Doron Haviv, Yaron Hadad, and Ido Kaminer. Generating conjectures on fundamental constants with the Ramanujan Machine. Nature, 590:67-73, 2021. URL:
  30. Helge Spieker and Arnaud Gotlieb. Learning objective boundaries for constraint optimization problems. In Giuseppe Nicosia, Varun Kumar Ojha, Emanuele La Malfa, Giorgio Jansen, Vincenzo Sciacca, Panos M. Pardalos, Giovanni Giuffrida, and Renato Umeton, editors, Machine Learning, Optimization, and Data Science - 6th International Conference, LOD 2020, Siena, Italy, July 19-23, 2020, Revised Selected Papers, Part II, volume 12566 of Lecture Notes in Computer Science, pages 394-408. Springer, 2020. URL:
  31. Ljupco Todorovski. Equation discovery. In Claude Sammut and Geoffrey I. Webb, editors, Encyclopedia of Machine Learning, pages 327-330. Springer, 2010. URL:
  32. Hao Wang. Toward mechanical mathematics. In Jörg Siekmann and Graham Wrightson, editors, Automation of Reasoning: Classical Papers on Computational Logic 1957-1966, pages 244-264. Springer-Verlag, Berlin, 1983. Google Scholar
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