UNO Gets Easier for a Single Player

This work is a follow up to [2, FUN 2010], which initiated a detailed analysis of the popular game of UNO R ©. We consider the solitaire version of the game, which was shown to be NP-complete. In [2], the authors also demonstrate a n ) 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.