Lemmings是PSPACE完备的
Lemmings is PSPACE-complete
摘要：
《旅鼠》（Lemmings）是一款由DMA Design开发并于1991年由Psygnosis出版的电脑益智游戏，玩家必须通过分配特定技能来改变他们的行为，从而引导一群旅鼠生物通过危险的景观到达安全地带。在本文中，我们研究的优化问题，保存最高数量的旅鼠在给定的景观与给定数量的可用技能。我们证明了游戏是PSPACE完备的，即使只有一只旅鼠需要拯救，并且只有建造者和破坏者技能可用。因此，我们解决了Cormode在2004年提出的一个开放问题，并在2010年再次由Forisek提出。然而，我们也证明，如果我们限制游戏的水平，其中可用的建设者技能只有多项式多（有任何数量的其他技能），那么游戏是NP可解的。类似地，如果可用的Basher、Miner和Digger技能是多项式多的，则游戏在NP中可解。此外，我们表明，保存最大数量的旅鼠是APX困难的，即使只有一种类型的技能，无论这种技能是什么。这与Cormode之前建立的决策问题的P中的成员限制在没有“致命区域”（如水或陷阱）的水平上，只有攀登者和漂浮者技能形成对比。
英文摘要：
Lemmings is a computer puzzle game developed by DMA Design and published by Psygnosis in 1991, in which the player has to guide a tribe of lemming creatures to safety through a hazardous landscape, by assigning them specific skills that modify their behavior in different ways. In this paper we study the optimization problem of saving the highest number of lemmings in a given landscape with a given number of available skills. We prove that the game is PSPACE-complete, even if there is only one lemming to save, and only Builder and Basher skills are available. We thereby settle an open problem posed by Cormode in 2004, and again by Forisek in 2010. However we also prove that, if we restrict the game to levels in which the available Builder skills are only polynomially many (and there is any number of other skills), then the game is solvable in NP. Similarly, if the available Basher, Miner, and Digger skills are polynomially many, the game is solvable in NP. Furthermore, we show that saving the maximum number of lemmings is APX-hard, even when only one type of skill is available, whatever this skill is. This contrasts with the membership in P of the decision problem restricted to levels with no "deadly areas" (such as water or traps) and only Climber and Floater skills, as previously established by Cormode.
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