Tetravex是NP完全的
Tetravex is NP-complete
摘要：
Tetravex是一个广泛玩的单人计算机游戏，在游戏中你会得到$n^2$单位瓷砖，每个瓷砖的边缘都标有一个数字。目标是将每个图块放置在$n$乘$n$的正方形内，使得所有相邻的边缘用相同的数字标记。不幸的是，玩Tetravex在计算上很难。更准确地说，我们证明，决定是否有一个平铺的Tetravex板是NP完全的。因此，决定在哪里放置瓷砖是NP困难的。这可能有助于解释为什么Tetravex是一个很好的拼图。这一结果补充了一些类似的结果，为一个人的游戏，涉及平铺。例如，NP-完全性结果已经被显示为：俄罗斯方块的离线版本，KPlumber（涉及旋转包含管道图纸的瓷砖以建立连接的网络）和最短滑动难题。它提出了一些悬而未决的问题。例如，无限版本是图灵完备的吗？我们如何生成真正令人困惑的Tetravex问题，因为随机NP完全问题通常令人惊讶地容易解决？我们能观察到相变行为吗？当保证有唯一解时，问题的复杂性又如何呢？我们如何生成具有唯一解的谜题？
英文摘要：
Tetravex is a widely played one person computer game in which you are given $n^2$ unit tiles, each edge of which is labelled with a number. The objective is to place each tile within a $n$ by $n$ square such that all neighbouring edges are labelled with an identical number. Unfortunately, playing Tetravex is computationally hard. More precisely, we prove that deciding if there is a tiling of the Tetravex board is NP-complete. Deciding where to place the tiles is therefore NP-hard. This may help to explain why Tetravex is a good puzzle. This result compliments a number of similar results for one person games involving tiling. For example, NP-completeness results have been shown for: the offline version of Tetris, KPlumber (which involves rotating tiles containing drawings of pipes to make a connected network), and shortest sliding puzzle problems. It raises a number of open questions. For example, is the infinite version Turing-complete? How do we generate Tetravex problems which are truly puzzling as random NP-complete problems are often surprising easy to solve? Can we observe phase transition behaviour? What about the complexity of the problem when it is guaranteed to have an unique solution? How do we generate puzzles with unique solutions?
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