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Complexity of Planar Graph Orientation Consistency, Promise-Inference, and Uniqueness, with Applications to Minesweeper Variants

Authors MIT Hardness Group, Della Hendrickson , Andy Tockman

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LIPIcs.FUN.2024.25.pdf
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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • NP
  • coNP
  • hardness
  • minesweeper
  • solvability
  • gadgets
  • simulation

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Abstract

We study three problems related to the computational complexity of the popular game Minesweeper. The first is consistency: given a set of clues, is there any arrangement of mines that satisfies it? This problem has been known to be NP-complete since 2000 [Kaye, 2000], but our framework proves it as a side effect. The second is inference: given a set of clues, is there any cell that the player can prove is safe? The coNP-completeness of this problem has been in the literature since 2011 [Scott et al., 2011], but we discovered a flaw that we believe is present in all published results, and we provide a fixed proof. Finally, the third is solvability: given the full state of a Minesweeper game, can the player win the game by safely clicking all non-mine cells? This problem has not yet been studied, and we prove that it is coNP-complete.

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MIT Hardness Group, Della Hendrickson, and Andy Tockman. Complexity of Planar Graph Orientation Consistency, Promise-Inference, and Uniqueness, with Applications to Minesweeper Variants. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FUN.2024.25

Author Details

MIT Hardness Group
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Della Hendrickson
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Andy Tockman
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Acknowledgements

This paper was initiated during open problem solving in the MIT class on Algorithmic Lower Bounds: Fun with Hardness Proofs (6.5440) taught by Erik Demaine in Fall 2023. We thank the other participants of that class for helpful discussions and providing an inspiring atmosphere.

References

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  7. Alex Thieme and Twan Basten. Minesweeper is difficult indeed! A Journey from Process Algebra via Timed Automata to Model Learning: Essays Dedicated to Frits Vaandrager on the Occasion of His 60th Birthday, 13560:472, 2022. Google Scholar
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