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Constraint Models for Klondike

Authors Nguyen Dang , Ian P. Gent , Peter Nightingale , Felix Ulrich-Oltean , Jack Waller

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Subject Classification

ACM Subject Classification
  • Theory of computation → Constraint and logic programming
  • Computing methodologies → Planning and scheduling
Keywords
  • AI Planning
  • Modelling
  • Constraint Programming
  • Solitaire and Patience Games

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Abstract

Klondike is the most famous single-player card game, and remains a challenging search problem even in the "thoughtful" variant where all card locations are known. We consider the full game of Klondike except for one restriction that the unusual move of "worrying back" is disallowed. This model is able to determine the winnability of all instances of the game and in practice does so in less than 2000 secs for 10,000 instances we tested, which no other known algorithm can achieve. On some instances, however, other techniques can produce answers more quickly. We use constraint modelling to produce schedules for running our constraint model in combination with other techniques. The combination outperforms any single solver across a range of time limits. Using this combination we are able to significantly improve the best estimate of winnability of Klondike without worrying back. Finally we show how we can use this work to also improve the estimate of winnability of the regular game of Klondike.

Cite As Get BibTex

Nguyen Dang, Ian P. Gent, Peter Nightingale, Felix Ulrich-Oltean, and Jack Waller. Constraint Models for Klondike. In 31st International Conference on Principles and Practice of Constraint Programming (CP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 340, pp. 9:1-9:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CP.2025.9

Author Details

Nguyen Dang
  • School of Computer Science, University of St Andrews, UK
Ian P. Gent
  • School of Computer Science, University of St Andrews, UK
Peter Nightingale
  • Department of Computer Science, University of York, UK
Felix Ulrich-Oltean
  • Department of Computer Science, University of York, UK
Jack Waller
  • Skyscanner, London, UK

Funding

  • Nightingale, Peter: EPSRC grant EP/W001977/1
  • Ulrich-Oltean, Felix: EPSRC grant EP/W001977/1

Acknowledgements

Experiments were carried out on the Viking and Cirrus clusters. The Viking cluster is a high performance computing facility provided by the University of York, and we are grateful for computational support from the University of York IT Services and the Research IT team. Cirrus is a UK National Tier-2 HPC Service at EPCC (http://www.cirrus.ac.uk) funded by the University of Edinburgh and EPSRC (EP/P020267/1). Thanks to reviewers for their careful comments.

Supplementary Materials

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