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A Constraint Programming Approach for n-Day Lookahead Playoff Clinching in the NHL

Authors Gili Rosenberg , Kyle E. C. Booth , J. Kyle Brubaker , Ruben S. Andrist

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Subject Classification

ACM Subject Classification
  • Theory of computation → Constraint and logic programming
  • Mathematics of computing → Combinatorial optimization
Keywords
  • Constraint programming
  • operations research
  • tree search
  • sports
  • hockey

Metrics

Abstract

In professional sports, a team has clinched the playoffs if they are guaranteed a postseason spot, regardless of the outcomes of any remaining games. As the season progresses, sports fans and other stakeholders are interested in precisely when, and under what conditions, their team will clinch the playoffs. In this paper, we investigate playoff clinching in the context of the National Hockey League (NHL), where it is computationally challenging to produce clinching scenarios due, in part, to complex tie-breakers. We present an algorithm that determines under which combinations of game outcomes in the next n days a team will clinch the playoffs (i.e., "n-day lookahead clinching"). Our approach is a custom tree search which employs various preprocessing techniques, pruning strategies, and node ordering heuristics to efficiently explore the space of possible outcomes. The tree search leverages a constraint programming (CP)-based subroutine for inference that determines if a team has clinched the playoffs for some snapshot in time of the regular season (i.e., "0-day lookahead clinching"). This CP subroutine aims to find a counter-example in which the team being evaluated is eliminated, taking into account qualification rules and the NHL’s extensive list of tie-breakers. We validate the efficacy of our algorithm using hundreds of scenarios based on public NHL data for the seasons 2021-22 through 2024-25. The methods introduced can be readily extended to other metrics of interest, including mathematical proof of playoff elimination, clinching the President’s Trophy, as well as clinching (or being eliminated from clinching) any other seed in the standings.

Cite As Get BibTex

Gili Rosenberg, Kyle E. C. Booth, J. Kyle Brubaker, and Ruben S. Andrist. A Constraint Programming Approach for n-Day Lookahead Playoff Clinching in the NHL. In 32nd International Conference on Principles and Practice of Constraint Programming (CP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 379, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CP.2026.48

Author Details

Gili Rosenberg
  • Amazon Advanced Solutions Lab, Seattle, WA, 98170, USA
Kyle E. C. Booth
  • Amazon Advanced Solutions Lab, Seattle, WA, 98170, USA
J. Kyle Brubaker
  • Amazon Advanced Solutions Lab, Seattle, WA, 98170, USA
Ruben S. Andrist
  • Amazon Advanced Solutions Lab, Seattle, WA, 98170, USA

Acknowledgements

The authors would like to acknowledge Helmut Katzgraber and Nino DiPasquale for their support, and Greg Inglis, Neil Pierson, Ben Resnick, and Katie Yates for productive discussions.

References

  1. Ilan Adler, Alan L Erera, Dorit S Hochbaum, and Eli V Olinick. Baseball, optimization, and the world wide web. Interfaces, 32(2):12-22, 2002. URL: https://doi.org/10.1287/inte.32.2.12.67.
  2. Thorsten Bernholt, Alexander Gülich, Thomas Hofmeister, and Niels Schmitt. Football elimination is hard to decide under the 3-point-rule. In International Symposium on Mathematical Foundations of Computer Science, pages 410-418. Springer, 1999. URL: https://doi.org/10.1007/3-540-48340-3_37.
  3. Diego Delle Donne and Javier Marenco. Studying playoff qualification in motorsports via mixed-integer programming techniques. Proceedings of the Institution of Mechanical Engineers, Part P: Journal of Sports Engineering and Technology, 226(1):32-41, 2012. Google Scholar
  4. Dan Gusfield and Charles Martel. The structure and complexity of sports elimination numbers. Algorithmica, 32(1):73-86, 2002. URL: https://doi.org/10.1007/s00453-001-0074-y.
  5. John Hooker. Logic-based methods for optimization: combining optimization and constraint satisfaction. John Wiley & Sons, 2011. Google Scholar
  6. Mark A Husted, Eli V Olinick, and Alexandra M Newman. Improving sports media’s crystal ball for national basketball association playoff elimination. INFORMS Journal on Applied Analytics, 51(2):119-135, 2021. URL: https://doi.org/10.1287/inte.2020.1034.
  7. Mark Andrew Husted. Enhancing Tractability of Mixed-Integer Nonlinear Programming Models: Case Studies in Energy and Sports. PhD thesis, Colorado School of Mines, 2019. Google Scholar
  8. Satoshi Ito and Yuji Shinano. Calculation of clinch and elimination numbers for sports leagues with multiple tiebreaking criteria. Technical Report ZIB-Report 18-51, Zuse Institute Berlin, 2018. Google Scholar
  9. Walter Kern and Daniël Paulusma. The computational complexity of the elimination problem in generalized sports competitions. Discrete Optimization, 1(2):205-214, 2004. URL: https://doi.org/10.1016/j.disopt.2003.12.003.
  10. Sam Sung Ho Kim, Mark A Husted, Eli V Olinick, and Alexandra M Newman. Improving South Korea’s crystal ball for baseball postseason clinching and elimination. INFORMS Journal on Applied Analytics, 54(4):298-311, 2024. URL: https://doi.org/10.1287/inte.2023.0035.
  11. Carlos JP Lucena, Thiago F Noronha, Celso C Ribeiro, and Sebastian Urrutia. A multi-agent framework to build integer programming applications to playoff elimination in sports tournaments. International Transactions in Operational Research, 15(6):739-753, 2008. URL: https://doi.org/10.1111/J.1475-3995.2008.00677.X.
  12. National Hockey League. NHL standings - tie-breaking procedure, 2025. Accessed: June 27, 2025. URL: https://www.nhl.com/info/standings-info/tie-breaking-procedure.
  13. National Hockey League. NHL Web API. https://api-web.nhle.com/, 2025. Accessed: 2025-05-05.
  14. NHL. Stanley cup playoff clinching scenarios for April 15, 2025. URL: https://www.nhl.com/news/stanley-cup-playoff-clinching-scenarios-for-april-15-2025.
  15. Laurent Perron and Frédéric Didier. CP-SAT v9.11.4210, 2024. URL: https://developers.google.com/optimization/cp/cp_solver/.
  16. Christian Raack, Annie Raymond, Thomas Schlechte, and Axel Werner. Standings in sports competitions using integer programming. Journal of Quantitative Analysis in Sports, 10(2):131-137, 2014. URL: https://doi.org/10.1515/jqas-2013-0111.
  17. Celso C Ribeiro and Sebastián Urrutia. An application of integer programming to playoff elimination in football championships. International Transactions in Operational Research, 12(4):375-386, 2005. URL: https://doi.org/10.1111/j.1475-3995.2005.00513.x.
  18. Lawrence W Robinson. Baseball playoff eliminations: An application of linear programming. Operations Research Letters, 10(2):67-74, 1991. URL: https://doi.org/10.1016/0167-6377(91)90089-8.
  19. Tyrel Russell. A computational study of problems in sports. PhD thesis, University of Waterloo, 2010. Google Scholar
  20. Tyrel Russell and Peter Van Beek. Mathematically clinching a playoff spot in the NHL and the effect of scoring systems. In Advances in Artificial Intelligence: 21st Conference of the Canadian Society for Computational Studies of Intelligence, Canadian AI 2008 Windsor, Canada, May 28-30, 2008 Proceedings, pages 234-245. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-68825-9_23.
  21. Tyrel Russell and Peter Van Beek. Determining the number of games needed to guarantee an NHL playoff spot. In International Conference on Integration of Constraint Programming, Artificial Intelligence, and Operations Research, pages 233-247. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-01929-6_18.
  22. Tyrel Russell and Peter Van Beek. Lessons learned from modelling the NHL playoff qualification problem. Constraint Modelling and Reformulation (ModRef'09), page 132, 2009. Google Scholar
  23. Tyrel Russell and Peter Van Beek. A hybrid constraint programming and enumeration approach for solving NHL playoff qualification and elimination problems. European Journal of Operational Research, 218(3):819-828, 2012. URL: https://doi.org/10.1016/j.ejor.2011.11.045.
  24. Benjamin L Schwartz. Possible winners in partially completed tournaments. SIAM Review, 8(3):302-308, 1966. URL: https://doi.org/10.1137/1008062.
  25. Kevin D Wayne. A new property and a faster algorithm for baseball elimination. SIAM Journal on Discrete Mathematics, 14(2):223-229, 2001. URL: https://doi.org/10.1137/S0895480198348847.
  26. Scott Whittle. The NFL true fan problem. PhD thesis, Kansas State University, 2014. Google Scholar
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