A SURVEY OF NP-COMPLETE PUZZLES Np-complete
字谜的研究综述
Graham Kendall Andrew Parkes Kristian Spoerer1
格雷厄姆·肯德尔·安德鲁·帕克斯克里斯蒂安·斯波尔
1 Nottingham, UK 英国诺丁汉
ABSTRACT 摘要
Single-player games (often called puzzles) have received considerable attention from the scientific community. Consequently, interesting insights into some puzzles, and into the approaches for solv-ing them, have emerged. However, many puzzles have been neglected, possibly because they are unknown to many people. In this article, we survey NP-Complete puzzles in the hope of motivat-ing further research in this fascinating area, particularly for those puzzles which have received little scientific attention to date.
单人游戏(通常称为拼图)受到了科学界的广泛关注。因此，对一些难题以及解决这 些难题的方法产生了有趣的见解。然而，很多谜题被忽视了，可能是因为很多人都 不知道。在这篇文章中，我们调查 np 完整的谜题，希望进一步的研究在这个迷人 的领域，特别是那些迄今为止还没有得到科学的重视的谜题。
- INTRODUCTION 引言
This article provides a survey of puzzles that are contained in the set of those NP-Complete (Garey and Johnson, 1979) puzzles that have a single decision maker. We use the term puzzle to refer to single-player games which are enjoyable to play. By definition a puzzle should have a solution which (1) is aesthetically pleasing and (2) gives the user satisfaction in reaching it. We admit that this is rather a loose definition. A more formal definition is difficult as any other definition would only express our subjective view and would be open to debate. Therefore, we decided upon the above definition which captures our perception of a single player puzzle. We provide further comments in the conclusions. Puzzles have been the topic of artificial intelligence research for some time (Robertson and Munro, 1978; Hearn, 2006) and this survey is intended to motivate further research in this area by providing a starting point for the interested researcher.
本文提供了一个包含在 NP-Complete(Garey 和 Johnson，1979)的集合中的谜题的调查，这些谜 题有一个决策者。我们用拼图这个术语来指单人游戏，玩起来很愉快。根据定义，一个难题 应该有一个解决方案，该方案(1)在美学上令人满意，(2)使用户在达到它时感到满意。我们承 认这是一个相当宽泛的定义。一个更正式的定义是困难的，因为任何其他定义只能表达我们 的主观观点，并将开放辩论。因此，我们决定采用上面的定义，这个定义抓住了我们对单人 游戏的理解。我们在结论中提供了进一步的意见。谜题一直是人工智能研究的主题(Robertson 和 Munro，1978;Hearn，2006)，这项调查的目的是通过为感兴趣的研究人员提供一个起点来 推动这一领域的进一步研究。
We discuss 24 puzzles presented in alphabetical order. Two puzzles (Rubikrsquo;s Cube and the Towers of Hanoi) are discussed separately. For each puzzle, a brief description of the rules and a reference to its complexity is provided. At the end of each puzzle description we summarize the decision problem that was used to prove its NP-Completeness. Note that some of the references are to URLs which suggests that there has been work directed towards these puzzles, but the work has not been presented for scientific peer review. It is our hope that this article will start to address the small number of published works, by encouraging authors to submit their research to journals such as this.
我们讨论了24 个字母顺序中提出的难题。分别讨论了魔方和河内塔这两个难题。对于每个难 题，都提供了规则的简要描述和对其复杂性的参考。在每个难题描述的最后，我们总结了用 于证明其 np 完全性的决策问题。需要注意的是，其中一些参考资料提到了 url，这表明已经 有针对这些难题的工作，但是这项工作还没有提交给科学同行评审。我们希望这篇文章能够 通过鼓励作者将他们的研究成果提交给这样的期刊，来解决少量已发表的作品。
A brief description of the puzzle is included so that the reader understands it. There is often a reference to some software so that the puzzle can be attempted which might give an appreciation of the subtleties involved. Furthermore, the reader might gain an insight into the similarities between these hard puzzles. For example, the implicit hidden constraints that are only revealed once a puzzle is attempted.
包括对这个难题的简要描述，以便读者能够理解。这里经常提到一些软件，这样就可以尝试 拼图，从而可以了解所涉及的微妙之处。此外，读者还可以深入了解这些难题之间的相似之 处。例如，只有在尝试拼图时才会显示的隐式隐藏约束。
A brief outline of some of the work that has been published about each puzzle is also provided. These works tend to apply search algorithms (see, for example, Russell and Norvig (2003)) to the puzzles. We have not attempted to include all of the research for each puzzle. Our motivation for briefly outlining this research is to provide references to the texts that can be used as starting points for future work on these puzzles.
简要概述了一些工作，已经发表了关于每一个难题也提供。这些工作倾向于将搜索算法(例如， 参见 Russell 和 Norvig(2003))应用于拼图。我们并没有试图包括每个难题的所有研究。我们简 要概述这项研究的动机是为了提供参考文献，可以作为未来在这些难题上工作的起点。
Erik Demaine has completed some work in the area of Combinatorial Games2 . We would like to recommend this work where it details the complexity of some two-player games and also some single-player puzzles. This work can be regarded as an extension of Demainersquo;s (2001) survey of puzzles, and also an extension of Eppsteinrsquo;s survey of puzzles3 . ErikDemaine
已经完成了组合游戏 2 领域的一些工作。我们想推荐这项工作，其中它详细说明 了一些双人游戏和一些单人游戏的复杂性。这项工作可以被看作是 Demaine(2001)对拼图的调 查的延伸，也是 Eppstein 对拼图调查的延伸。
2. COMPUTATIONAL COMPLEXITY
2。计算复杂性
In this section, we provide an informal description of NP-Completeness. The treatment is informal by design as there are many other more formal treatments of this area, with Garey and Johnson (1979) being the classic text. For the interested reader we refer to the seminal work by Stephen Cook (1971) and also his short article (Cook, 1984) which provides an excellent discussion. In fact, the opening of the article succinctly shows the problem faced when considering computational complexity: “In general, it is much harder to find a solution to a problem than to recognize one when it is presented.”
在本节中，我们将提供一个对 np 完备性的非正式描述。治疗是非正式的设计，因为有许多其 他更正式的治疗这一领域，与 Garey 和约翰逊(1979 年)是经典的文本。对于感兴趣的读者， 我们参考史蒂芬库克(1971 年)的开创性工作，以及他的短文(库克，1984 年)，其中提供了一 个极好的讨论。事实上，文章的开头简明扼要地说明了在考虑计算复杂性时所面临的问题:'一 般来说，找到问题的解决方案比在提出问题时识别它要困难得多。
' We are concerned with decision problems. That is, those problems that can be answered by Yes or No. This might appear to be quite a restriction on the type of problems we can consider but, in fact, most problems can be reduced to a decision problem. For example, the intuitive way to consider the Travelling Salesman Problem (TSP) is to phrase it as “Given a TSP instance,
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A SURVEY OF NP-COMPLETE PUZZLES Np-complete
字谜的研究综述
Graham Kendall Andrew Parkes Kristian Spoerer1
格雷厄姆·肯德尔·安德鲁·帕克斯克里斯蒂安·斯波尔
1 Nottingham, UK 英国诺丁汉
ABSTRACT 摘要
Single-player games (often called puzzles) have received considerable attention from the scientific community. Consequently, interesting insights into some puzzles, and into the approaches for solv-ing them, have emerged. However, many puzzles have been neglected, possibly because they are unknown to many people. In this article, we survey NP-Complete puzzles in the hope of motivat-ing further research in this fascinating area, particularly for those puzzles which have received little scientific attention to date.
单人游戏(通常称为拼图)受到了科学界的广泛关注。因此，对一些难题以及解决这 些难题的方法产生了有趣的见解。然而，很多谜题被忽视了，可能是因为很多人都 不知道。在这篇文章中，我们调查 np 完整的谜题，希望进一步的研究在这个迷人 的领域，特别是那些迄今为止还没有得到科学的重视的谜题。
- INTRODUCTION 引言
This article provides a survey of puzzles that are contained in the set of those NP-Complete (Garey and Johnson, 1979) puzzles that have a single decision maker. We use the term puzzle to refer to single-player games which are enjoyable to play. By definition a puzzle should have a solution which (1) is aesthetically pleasing and (2) gives the user satisfaction in reaching it. We admit that this is rather a loose definition. A more formal definition is difficult as any other definition would only express our subjective view and would be open to debate. Therefore, we decided upon the above definition which captures our perception of a single player puzzle. We provide further comments in the conclusions. Puzzles have been the topic of artificial intelligence research for some time (Robertson and Munro, 1978; Hearn, 2006) and this survey is intended to motivate further research in this area by providing a starting point for the interested researcher.
本文提供了一个包含在 NP-Complete(Garey 和 Johnson，1979)的集合中的谜题的调查，这些谜 题有一个决策者。我们用拼图这个术语来指单人游戏，玩起来很愉快。根据定义，一个难题 应该有一个解决方案，该方案(1)在美学上令人满意，(2)使用户在达到它时感到满意。我们承 认这是一个相当宽泛的定义。一个更正式的定义是困难的，因为任何其他定义只能表达我们 的主观观点，并将开放辩论。因此，我们决定采用上面的定义，这个定义抓住了我们对单人 游戏的理解。我们在结论中提供了进一步的意见。谜题一直是人工智能研究的主题(Robertson 和 Munro，1978;Hearn，2006)，这项调查的目的是通过为感兴趣的研究人员提供一个起点来 推动这一领域的进一步研究。
We discuss 24 puzzles presented in alphabetical order. Two puzzles (Rubikrsquo;s Cube and the Towers of Hanoi) are discussed separately. For each puzzle, a brief description of the rules and a reference to its complexity is provided. At the end of each puzzle description we summarize the decision problem that was used to prove its NP-Completeness. Note that some of the references are to URLs which suggests that there has been work directed towards these puzzles, but the work has not been presented for scientific peer review. It is our hope that this article will start to address the small number of published works, by encouraging authors to submit their research to journals such as this.
我们讨论了24 个字母顺序中提出的难题。分别讨论了魔方和河内塔这两个难题。对于每个难 题，都提供了规则的简要描述和对其复杂性的参考。在每个难题描述的最后，我们总结了用 于证明其 np 完全性的决策问题。需要注意的是，其中一些参考资料提到了 url，这表明已经 有针对这些难题的工作，但是这项工作还没有提交给科学同行评审。我们希望这篇文章能够 通过鼓励作者将他们的研究成果提交给这样的期刊，来解决少量已发表的作品。
A brief description of the puzzle is included so that the reader understands it. There is often a reference to some software so that the puzzle can be attempted which might give an appreciation of the subtleties involved. Furthermore, the reader might gain an insight into the similarities between these hard puzzles. For example, the implicit hidden constraints that are only revealed once a puzzle is attempted.
包括对这个难题的简要描述，以便读者能够理解。这里经常提到一些软件，这样就可以尝试 拼图，从而可以了解所涉及的微妙之处。此外，读者还可以深入了解这些难题之间的相似之 处。例如，只有在尝试拼图时才会显示的隐式隐藏约束。
A brief outline of some of the work that has been published about each puzzle is also provided. These works tend to apply search algorithms (see, for example, Russell and Norvig (2003)) to the puzzles. We have not attempted to include all of the research for each puzzle. Our motivation for briefly outlining this research is to provide references to the texts that can be used as starting points for future work on these puzzles.
简要概述了一些工作，已经发表了关于每一个难题也提供。这些工作倾向于将搜索算法(例如， 参见 Russell 和 Norvig(2003))应用于拼图。我们并没有试图包括每个难题的所有研究。我们简 要概述这项研究的动机是为了提供参考文献，可以作为未来在这些难题上工作的起点。
Erik Demaine has completed some work in the area of Combinatorial Games2 . We would like to recommend this work where it details the complexity of some two-player games and also some single-player puzzles. This work can be regarded as an extension of Demainersquo;s (2001) survey of puzzles, and also an extension of Eppsteinrsquo;s survey of puzzles3 . ErikDemaine
已经完成了组合游戏 2 领域的一些工作。我们想推荐这项工作，其中它详细说明 了一些双人游戏和一些单人游戏的复杂性。这项工作可以被看作是 Demaine(2001)对拼图的调 查的延伸，也是 Eppstein 对拼图调查的延伸。
2. COMPUTATIONAL COMPLEXITY
2。计算复杂性
In this section, we provide an informal description of NP-Completeness. The treatment is informal by design as there are many other more formal treatments of this area, with Garey and Johnson (1979) being the classic text. For the interested reader we refer to the seminal work by Stephen Cook (1971) and also his short article (Cook, 1984) which provides an excellent discussion. In fact, the opening of the article succinctly shows the problem faced when considering computational complexity: “In general, it is much harder to find a solution to a problem than to recognize one when it is presented.”
在本节中，我们将提供一个对 np 完备性的非正式描述。治疗是非正式的设计，因为有许多其 他更正式的治疗这一领域，与 Garey 和约翰逊(1979 年)是经典的文本。对于感兴趣的读者， 我们参考史蒂芬库克(1971 年)的开创性工作，以及他的短文(库克，1984 年)，其中提供了一 个极好的讨论。事实上，文章的开头简明扼要地说明了在考虑计算复杂性时所面临的问题:'一 般来说，找到问题的解决方案比在提出问题时识别它要困难得多。
' We are concerned with decision problems. That is, those problems that can be answered by Yes or No. This might appear to be quite a restriction on the type of problems we can consider but, in fact, most problems can be reduced to a decision problem. For example, the intuitive way to consider the Travelling Salesman Problem (TSP) is to phrase it as “Given a TSP instance,
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