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UNO Gets Easier for a Single Player

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8496)

Abstract

This work is a follow up to[2, FUN 2010], which initiated a detailed analysis of the popular game of UNO®. We consider the solitaire version of the game, which was shown to be NP-complete. In[2], the authors also demonstrate a \(n^{O(c^2)}\) algorithm, where c is the number of colors across all the cards, which implies, in particular that the problem is polynomial time when the number of colors is a constant.

In this work, we propose a kernelization algorithm, a consequence of which is that the problem is fixed-parameter tractable when the number of colors is treated as a parameter. This removes the exponential dependence on c and answers the question stated in[2] in the affirmative. We also introduce a natural and possibly more challenging version of UNO that we call “All Or None UNO”. For this variant, we prove that even the single-player version is NP-complete, and we show a single-exponential FPT algorithm, along with a cubic kernel.

Keywords

  • Reduction Rule
  • Feasible Sequence
  • Label Pair
  • Popular Game
  • Maximal Subsequence

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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  • DOI: 10.1007/978-3-319-07890-8_13
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References

  1. Alon, Yuster, Zwick: Color-Coding. JACM: Journal of the ACM 42 (1995)

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  2. Demaine, E.D., Demaine, M.L., Uehara, R., Uno, T., Uno, Y.: UNO Is Hard, Even for a Single Player. In: Boldi, P. (ed.) FUN 2010. LNCS, vol. 6099, pp. 133–144. Springer, Heidelberg (2010)

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© 2014 Springer International Publishing Switzerland

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Dey, P., Goyal, P., Misra, N. (2014). UNO Gets Easier for a Single Player. In: Ferro, A., Luccio, F., Widmayer, P. (eds) Fun with Algorithms. FUN 2014. Lecture Notes in Computer Science, vol 8496. Springer, Cham. https://doi.org/10.1007/978-3-319-07890-8_13

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  • DOI: https://doi.org/10.1007/978-3-319-07890-8_13

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-07889-2

  • Online ISBN: 978-3-319-07890-8

  • eBook Packages: Computer ScienceComputer Science (R0)