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Hamiltonian Paths and Cycles in NP-Complete Puzzles

Authors Marnix Deurloo , Mitchell Donkers, Mieke Maarse, Benjamin G. Rin , Karen Schutte

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Hamiltonicity
  • NP-completeness
  • complexity theory
  • pen-and-paper puzzles

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Abstract

We show that several pen-and-paper puzzles are NP-complete by giving polynomial-time reductions from the Hamiltonian path and Hamiltonian cycle problems on grid graphs with maximum degree 3. The puzzles include Dotchi Loop, Chains, Linesweeper, Arukone{}₃ (also called Numberlink₃), and Araf. In addition, we show that this type of proof can still be used to prove the NP-completeness of Dotchi Loop even when the available puzzle instances are heavily restricted. Together, these results suggest that this approach holds promise in general for finding NP-completeness proofs of many pen-and-paper puzzles.

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Marnix Deurloo, Mitchell Donkers, Mieke Maarse, Benjamin G. Rin, and Karen Schutte. Hamiltonian Paths and Cycles in NP-Complete Puzzles. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 11:1-11:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FUN.2024.11

Author Details

Marnix Deurloo
  • Utrecht University, The Netherlands
Mitchell Donkers
  • Utrecht University, The Netherlands
Mieke Maarse
  • Utrecht University, The Netherlands
Benjamin G. Rin
  • Utrecht University, The Netherlands
Karen Schutte
  • Utrecht University, The Netherlands

Acknowledgements

We would like to thank Hein Duijf, Jonathan Grube, Rosalie Iemhoff, Dominik Klein, Michael Moortgat, and the anonymous referees for their helpful comments and suggestions.

References

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