Document Open Access Logo

The Distance Constraint on Sequence Variables

Authors Margaux Schmied , Augustin Delecluse , Jean-Charles Régin , Pierre Schaus

PDF

File

Thumbnail PDF
PDF
LIPIcs.CP.2026.49.pdf
  • Filesize: 1.82 MB
  • 19 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Constraint and logic programming
Keywords
  • Constraint programming
  • sequence variable
  • global constraint
  • cost-based constraint
  • distance constraint
  • lower bound

Metrics

Abstract

Insertion sequence variables have recently been introduced as a computational domain for modeling routing and sequencing problems in constraint programming. Typically, search heuristics guide the insertion process of new nodes into a partial growing path, while constraints eliminate infeasible insertions. This paper investigates filtering for the (minimum) distance constraint over insertion sequence variables. This global constraint links a sequence to a distance variable based on a given distance matrix. So far, only a simple filtering algorithm has been proposed, which considers the partial path but ignores mandatory nodes. Our contribution is to introduce stronger lower bounds that also take mandatory nodes into account. These bounds further enable the derivation of additional filtering rules for node insertions. An experimental evaluation on the TourMustSee problem shows that the proposed filtering rules significantly reduce the search space compared to the existing filtering approach.

Cite As Get BibTex

Margaux Schmied, Augustin Delecluse, Jean-Charles Régin, and Pierre Schaus. The Distance Constraint on Sequence Variables. In 32nd International Conference on Principles and Practice of Constraint Programming (CP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 379, pp. 49:1-49:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CP.2026.49

Author Details

Margaux Schmied
  • Université Côte d'Azur, France
Augustin Delecluse
  • KU Leuven, Belgium
Jean-Charles Régin
  • Université Côte d'Azur, France
Pierre Schaus
  • UCLouvain, Louvain-la-Neuve, Belgium

Funding

This work has been supported by the French government, through the 3IA Côte d’Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002.

References

  1. Richard Bellman. On a routing problem. Quarterly of Applied Mathematics, 16(1):87-90, 1958. URL: http://www.jstor.org/stable/43634538.
  2. Pascal Benchimol, Jean-Charles Régin, Louis-Martin Rousseau, Michel Rueher, and Willem-Jan Van Hoeve. Improving the held and karp approach with constraint programming. In International Conference on Integration of Artificial Intelligence (AI) and Operations Research (OR) Techniques in Constraint Programming, pages 40-44. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-13520-0_6.
  3. Geoffrey Chu, Peter J. Stuckey, Andreas Schutt, Thorsten Ehlers, Graeme Gange, and Kathryn Francis. Chuffed - A lazy clause solver, 2016. URL: https://github.com/chuffed/chuffed.
  4. Goerge Dantzig, Richard Fulkerson, and Simon Johnson. Solution of a large-scale traveling-salesman problem. Journal of the Operations Research Society of America, 2(4):393-410, 1954. URL: https://doi.org/10.1287/OPRE.2.4.393.
  5. Augustin Delecluse, Pierre Schaus, and Pascal Van Hentenryck. Sequence variables for routing problems. In 28th International Conference on Principles and Practice of Constraint Programming (CP 2022), pages 19:1-19:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CP.2022.19.
  6. Augustin Delecluse, Pierre Schaus, and Pascal Van Hentenryck. Sequence variables: A constraint programming computational domain for routing and sequencing. Manuscript in preparation, 2025. Google Scholar
  7. Frédéric Didier, Laurent Perron, Sarah Mohajeri, Steven Agostino Gay, Thibaut Cuvelier, and Vincent Furnon. Or-tools’ vehicle routing solver: A generic constraint-programming solver with heuristic search for routing problems, 2023. Google Scholar
  8. Jean-Guillaume Fages, Xavier Lorca, and Louis-Martin Rousseau. The salesman and the tree: the importance of search in cp. Constraints, 21(2):145-162, 2016. URL: https://doi.org/10.1007/S10601-014-9178-2.
  9. Filippo Focacci, Andrea Lodi, and Michela Milano. Cost-based domain filtering. In International conference on principles and practice of constraint programming, pages 189-203. Springer, 1999. URL: https://doi.org/10.1007/978-3-540-48085-3_14.
  10. Filippo Focacci, Andrea Lodi, and Michela Milano. Embedding relaxations in global constraints for solving tsp and tsptw. Ann. Math. Artif. Intell., 34:291-311, April 2002. URL: https://doi.org/10.1023/A:1014492408220.
  11. Lester R Ford Jr. Network flow theory. Technical report, RAND Corporation, 1956. Google Scholar
  12. Lester R Ford Jr and Delbert R Fulkerson. Maximal flow through a network. Canadian journal of Mathematics, 8:399-404, 1956. Google Scholar
  13. John Forrest, Ted Ralphs, Stefan Vigerske, Haroldo Gambini Santos, et al. COIN-OR Branch-and-Cut (Cbc), 2021. COIN-OR Branch-and-Cut (CBC) mixed integer optimizer. URL: https://doi.org/10.5281/zenodo.2720283.
  14. Michael Held and Richard M. Karp. The traveling-salesman problem and minimum spanning trees. Operations Research, 18(6):1138-1162, 1970. URL: https://doi.org/10.1287/OPRE.18.6.1138.
  15. Nicolas Isoart and Jean-Charles Régin. A linear time algorithm for the k-cutset constraint. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021), pages 29:1-29:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.CP.2021.29.
  16. Mustafa Küçük and Seyda Topaloglu Yildiz. Constraint programming-based solution approaches for three-dimensional loading capacitated vehicle routing problems. Computers & Industrial Engineering, 171:108505, 2022. URL: https://doi.org/10.1016/J.CIE.2022.108505.
  17. Kwan Hui Lim, Jeffrey Chan, Christopher Leckie, and Shanika Karunasekera. Personalized trip recommendation for tourists based on user interests, points of interest visit durations and visit recency. Knowledge and Information Systems, 54(2):375-406, 2018. URL: https://doi.org/10.1007/S10115-017-1056-Y.
  18. Clair E Miller, Albert W Tucker, and Richard A Zemlin. Integer programming formulation of traveling salesman problems. Journal of the ACM (JACM), 7(4):326-329, 1960. URL: https://doi.org/10.1145/321043.321046.
  19. Roberto Montemanni and Mauro Dell'Amico. Constraint programming models for the parallel drone scheduling vehicle routing problem. EURO Journal on Computational Optimization, 11:100078, 2023. URL: https://doi.org/10.1016/J.EJCO.2023.100078.
  20. Nicholas Nethercote, Peter J Stuckey, Ralph Becket, Sebastian Brand, Gregory J Duck, and Guido Tack. MiniZinc: Towards a standard CP modelling language. In International Conference on Principles and Practice of Constraint Programming, pages 529-543. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-74970-7_38.
  21. Augustin Parjadis, Quentin Cappart, Bistra Dilkina, Aaron Ferber, and Louis-Martin Rousseau. Learning lagrangian multipliers for the travelling salesman problem. In 30th International Conference on Principles and Practice of Constraint Programming, 2024. Google Scholar
  22. Laurent Perron, Frédéric Didier, and Steven Gay. The cp-sat-lp solver (invited talk). In 29th International Conference on Principles and Practice of Constraint Programming (CP 2023), pages 3:1-3:2. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.CP.2023.3.
  23. Gilles Pesant, Michel Gendreau, Jean-Yves Potvin, and Jean-Marc Rousseau. An exact constraint logic programming algorithm for the traveling salesman problem with time windows. Transportation Science, 32(1):12-29, 1998. URL: https://doi.org/10.1287/TRSC.32.1.12.
  24. Gilles Pesant, Michel Gendreau, Jean-Yves Potvin, and Jean-Marc Rousseau. On the flexibility of constraint programming models: From single to multiple time windows for the traveling salesman problem. European Journal of Operational Research, 117(2):253-263, 1999. URL: https://doi.org/10.1016/S0377-2217(98)00248-3.
  25. Jean-Charles Régin. A filtering algorithm for constraints of difference in csps. In Proceedings of the Twelfth AAAI National Conference on Artificial Intelligence, pages 362-367, 1994. URL: http://www.aaai.org/Library/AAAI/1994/aaai94-055.php.
  26. Jean-Charles Régin. Cost-based arc consistency for global cardinality constraints. Constraints, 7(3/4):387-405, July 2002. URL: https://doi.org/10.1023/A:1020506526052.
  27. Pierre Schaus, Guillaume Derval, Augustin Delecluse, Laurent Michel, and Pascal Van Hentenryck. Maxicp: A constraint programming solver for scheduling and vehicle routing, 2024. URL: https://github.com/aia-uclouvain/maxicp.
  28. Margaux Schmied and Jean-Charles Régin. Efficient implementation of the global cardinality constraint with costs. In 30th International Conference on Principles and Practice of Constraint Programming (CP 2024), pages 27:1-27:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.CP.2024.27.
  29. Christian Schulte, Guido Tack, and Mikael Z Lagerkvist. Modeling and programming with gecode. Schulte, Christian and Tack, Guido and Lagerkvist, Mikael, 1, 2010. Google Scholar
  30. Ole Tange. Gnu parallel - The command-line power tool. ;login: The USENIX Magazine, 36(1):42-47, February 2011. URL: http://www.gnu.org/s/parallel.
  31. Kendall Taylor, Kwan Hui Lim, and Jeffrey Chan. Travel itinerary recommendations with must-see points-of-interest. In Companion proceedings of the the web conference 2018, pages 1198-1205, 2018. URL: https://doi.org/10.1145/3184558.3191558.
  32. Charles Thomas, Roger Kameugne, and Pierre Schaus. Insertion sequence variables for hybrid routing and scheduling problems. In International Conference on Integration of Constraint Programming, Artificial Intelligence, and Operations Research, pages 457-474. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-58942-4_30.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact