UNO Gets Easier for a Single Player

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.