条条大路通罗马
All Paths Lead to Rome
摘要：
条条大路通罗马是罗马益智游戏的核心思想。它在由方形单元组成的$n \times n$网格上播放。这些单元格被分组到最多四个相邻单元格的框中，并且用指向基本方向的箭头填充或待填充。游戏的目标是用箭头填充空的单元格，这样每个盒子最多包含一个方向的箭头，无论我们从哪里开始，如果我们沿着单元格中的箭头，我们总是会在特殊的罗马单元格中结束。在这项工作中，我们研究了计算复杂性的益智游戏罗马，并表明，完成罗马董事会根据规则是一个\NP -完成的任务，计数的有效完成的数量是#Ptime-complete，并确定预先设置的箭头的数量需要使实例\emph{uniquely}可解是$Σ_2^P$ -完成。进一步证明了在ETH环境下，在$n\times n$板上完成给定Roma实例的问题不能在$\mathcal{O}\left(2^{{o}(n)}\right)$中及时解决，并基于Catalan结构的思想给出了一个匹配的动态规划算法.
英文摘要：
All roads lead to Rome is the core idea of the puzzle game Roma. It is played on an $n \times n$ grid consisting of quadratic cells. Those cells are grouped into boxes of at most four neighboring cells and are either filled, or to be filled, with arrows pointing in cardinal directions. The goal of the game is to fill the empty cells with arrows such that each box contains at most one arrow of each direction and regardless where we start, if we follow the arrows in the cells, we will always end up in the special Roma-cell. In this work, we study the computational complexity of the puzzle game Roma and show that completing a Roma board according to the rules is an \NP-complete task, counting the number of valid completions is #Ptime-complete, and determining the number of preset arrows needed to make the instance \emph{uniquely} solvable is $Σ_2^P$-complete. We further show that the problem of completing a given Roma instance on an $n\times n$ board cannot be solved in time $\mathcal{O}\left(2^{{o}(n)}\right)$ under ETH and give a matching dynamic programming algorithm based on the idea of Catalan structures.
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